def quad(func, a, b, args=(), full_output=<NUM_LIT:0>, epsabs=<NUM_LIT>, epsrel=<NUM_LIT>,<EOL>limit=<NUM_LIT:50>, points=None, weight=None, wvar=None, wopts=None, maxp1=<NUM_LIT:50>,<EOL>limlst=<NUM_LIT:50>):
if not isinstance(args, tuple):<EOL><INDENT>args = (args,)<EOL><DEDENT>if (weight is None):<EOL><INDENT>retval = _quad(func, a, b, args, full_output, epsabs, epsrel, limit,<EOL>points)<EOL><DEDENT>else:<EOL><INDENT>retval = _quad_weight(func, a, b, args, full_output, epsabs, epsrel,<EOL>limlst, limit, maxp1, weight, wvar, wopts)<EOL><DEDENT>ier = retval[-<NUM_LIT:1>]<EOL>if ier == <NUM_LIT:0>:<EOL><INDENT>return retval[:-<NUM_LIT:1>]<EOL><DEDENT>msgs = {<NUM_LIT>: "<STR_LIT>",<EOL><NUM_LIT:1>: "<STR_LIT>" % limit,<EOL><NUM_LIT:2>: "<STR_LIT>",<EOL><NUM_LIT:3>: "<STR_LIT>",<EOL><NUM_LIT:4>: "<STR_LIT>",<EOL><NUM_LIT:5>: "<STR_LIT>",<EOL><NUM_LIT:6>: "<STR_LIT>",<EOL><NUM_LIT:7>: "<STR_LIT>",<EOL>'<STR_LIT>': "<STR_LIT>"}<EOL>if weight in ['<STR_LIT>','<STR_LIT>'] and (b == Inf or a == -Inf):<EOL><INDENT>msgs[<NUM_LIT:1>] = "<STR_LIT>"<EOL>msgs[<NUM_LIT:4>] = "<STR_LIT>"<EOL>msgs[<NUM_LIT:7>] = "<STR_LIT>"<EOL>explain = {<NUM_LIT:1>: "<STR_LIT>",<EOL><NUM_LIT:2>: "<STR_LIT>",<EOL><NUM_LIT:3>: "<STR_LIT>",<EOL><NUM_LIT:4>: "<STR_LIT>",<EOL><NUM_LIT:5>: "<STR_LIT>"}<EOL><DEDENT>try:<EOL><INDENT>msg = msgs[ier]<EOL><DEDENT>except KeyError:<EOL><INDENT>msg = msgs['<STR_LIT>']<EOL><DEDENT>if ier in [<NUM_LIT:1>,<NUM_LIT:2>,<NUM_LIT:3>,<NUM_LIT:4>,<NUM_LIT:5>,<NUM_LIT:7>]:<EOL><INDENT>if full_output:<EOL><INDENT>if weight in ['<STR_LIT>','<STR_LIT>'] and (b == Inf or a == Inf):<EOL><INDENT>return retval[:-<NUM_LIT:1>] + (msg, explain)<EOL><DEDENT>else:<EOL><INDENT>return retval[:-<NUM_LIT:1>] + (msg,)<EOL><DEDENT><DEDENT>else:<EOL><INDENT>warnings.warn(msg, IntegrationWarning)<EOL>return retval[:-<NUM_LIT:1>]<EOL><DEDENT><DEDENT>else:<EOL><INDENT>raise ValueError(msg)<EOL><DEDENT>
Compute a definite integral.

Integrate func from `a` to `b` (possibly infinite interval) using a
technique from the Fortran library QUADPACK.

Parameters
----------
func : function
    A Python function or method to integrate.  If `func` takes many
    arguments, it is integrated along the axis corresponding to the
    first argument.
    If the user desires improved integration performance, then f may
    instead be a ``ctypes`` function of the form:

        f(int n, double args[n]),

    where ``args`` is an array of function arguments and ``n`` is the
    length of ``args``. ``f.argtypes`` should be set to
    ``(c_int, c_double)``, and ``f.restype`` should be ``(c_double,)``.
a : float
    Lower limit of integration (use -numpy.inf for -infinity).
b : float
    Upper limit of integration (use numpy.inf for +infinity).
args : tuple, optional
    Extra arguments to pass to `func`.
full_output : int, optional
    Non-zero to return a dictionary of integration information.
    If non-zero, warning messages are also suppressed and the
    message is appended to the output tuple.

Returns
-------
y : float
    The integral of func from `a` to `b`.
abserr : float
    An estimate of the absolute error in the result.
infodict : dict
    A dictionary containing additional information.
    Run scipy.integrate.quad_explain() for more information.
message :
    A convergence message.
explain :
    Appended only with 'cos' or 'sin' weighting and infinite
    integration limits, it contains an explanation of the codes in
    infodict['ierlst']

Other Parameters
----------------
epsabs : float or int, optional
    Absolute error tolerance.
epsrel : float or int, optional
    Relative error tolerance.
limit : float or int, optional
    An upper bound on the number of subintervals used in the adaptive
    algorithm.
points : (sequence of floats,ints), optional
    A sequence of break points in the bounded integration interval
    where local difficulties of the integrand may occur (e.g.,
    singularities, discontinuities). The sequence does not have
    to be sorted.
weight : float or int, optional
    String indicating weighting function. Full explanation for this
    and the remaining arguments can be found below.
wvar : optional
    Variables for use with weighting functions.
wopts : optional
    Optional input for reusing Chebyshev moments.
maxp1 : float or int, optional
    An upper bound on the number of Chebyshev moments.
limlst : int, optional
    Upper bound on the number of cycles (>=3) for use with a sinusoidal
    weighting and an infinite end-point.

See Also
--------
dblquad : double integral
tplquad : triple integral
nquad : n-dimensional integrals (uses `quad` recursively)
fixed_quad : fixed-order Gaussian quadrature
quadrature : adaptive Gaussian quadrature
odeint : ODE integrator
ode : ODE integrator
simps : integrator for sampled data
romb : integrator for sampled data
scipy.special : for coefficients and roots of orthogonal polynomials

Notes
-----

**Extra information for quad() inputs and outputs**

If full_output is non-zero, then the third output argument
(infodict) is a dictionary with entries as tabulated below.  For
infinite limits, the range is transformed to (0,1) and the
optional outputs are given with respect to this transformed range.
Let M be the input argument limit and let K be infodict['last'].
The entries are:

'neval'
    The number of function evaluations.
'last'
    The number, K, of subintervals produced in the subdivision process.
'alist'
    A rank-1 array of length M, the first K elements of which are the
    left end points of the subintervals in the partition of the
    integration range.
'blist'
    A rank-1 array of length M, the first K elements of which are the
    right end points of the subintervals.
'rlist'
    A rank-1 array of length M, the first K elements of which are the
    integral approximations on the subintervals.
'elist'
    A rank-1 array of length M, the first K elements of which are the
    moduli of the absolute error estimates on the subintervals.
'iord'
    A rank-1 integer array of length M, the first L elements of
    which are pointers to the error estimates over the subintervals
    with ``L=K`` if ``K<=M/2+2`` or ``L=M+1-K`` otherwise. Let I be the
    sequence ``infodict['iord']`` and let E be the sequence
    ``infodict['elist']``.  Then ``E[I[1]], ..., E[I[L]]`` forms a
    decreasing sequence.

If the input argument points is provided (i.e. it is not None),
the following additional outputs are placed in the output
dictionary.  Assume the points sequence is of length P.

'pts'
    A rank-1 array of length P+2 containing the integration limits
    and the break points of the intervals in ascending order.
    This is an array giving the subintervals over which integration
    will occur.
'level'
    A rank-1 integer array of length M (=limit), containing the
    subdivision levels of the subintervals, i.e., if (aa,bb) is a
    subinterval of ``(pts[1], pts[2])`` where ``pts[0]`` and ``pts[2]``
    are adjacent elements of ``infodict['pts']``, then (aa,bb) has level l
    if ``|bb-aa| = |pts[2]-pts[1]| * 2**(-l)``.
'ndin'
    A rank-1 integer array of length P+2.  After the first integration
    over the intervals (pts[1], pts[2]), the error estimates over some
    of the intervals may have been increased artificially in order to
    put their subdivision forward.  This array has ones in slots
    corresponding to the subintervals for which this happens.

**Weighting the integrand**

The input variables, *weight* and *wvar*, are used to weight the
integrand by a select list of functions.  Different integration
methods are used to compute the integral with these weighting
functions.  The possible values of weight and the corresponding
weighting functions are.

==========  ===================================   =====================
``weight``  Weight function used                  ``wvar``
==========  ===================================   =====================
'cos'       cos(w*x)                              wvar = w
'sin'       sin(w*x)                              wvar = w
'alg'       g(x) = ((x-a)**alpha)*((b-x)**beta)   wvar = (alpha, beta)
'alg-loga'  g(x)*log(x-a)                         wvar = (alpha, beta)
'alg-logb'  g(x)*log(b-x)                         wvar = (alpha, beta)
'alg-log'   g(x)*log(x-a)*log(b-x)                wvar = (alpha, beta)
'cauchy'    1/(x-c)                               wvar = c
==========  ===================================   =====================

wvar holds the parameter w, (alpha, beta), or c depending on the weight
selected.  In these expressions, a and b are the integration limits.

For the 'cos' and 'sin' weighting, additional inputs and outputs are
available.

For finite integration limits, the integration is performed using a
Clenshaw-Curtis method which uses Chebyshev moments.  For repeated
calculations, these moments are saved in the output dictionary:

'momcom'
    The maximum level of Chebyshev moments that have been computed,
    i.e., if ``M_c`` is ``infodict['momcom']`` then the moments have been
    computed for intervals of length ``|b-a| * 2**(-l)``,
    ``l=0,1,...,M_c``.
'nnlog'
    A rank-1 integer array of length M(=limit), containing the
    subdivision levels of the subintervals, i.e., an element of this
    array is equal to l if the corresponding subinterval is
    ``|b-a|* 2**(-l)``.
'chebmo'
    A rank-2 array of shape (25, maxp1) containing the computed
    Chebyshev moments.  These can be passed on to an integration
    over the same interval by passing this array as the second
    element of the sequence wopts and passing infodict['momcom'] as
    the first element.

If one of the integration limits is infinite, then a Fourier integral is
computed (assuming w neq 0).  If full_output is 1 and a numerical error
is encountered, besides the error message attached to the output tuple,
a dictionary is also appended to the output tuple which translates the
error codes in the array ``info['ierlst']`` to English messages.  The
output information dictionary contains the following entries instead of
'last', 'alist', 'blist', 'rlist', and 'elist':

'lst'
    The number of subintervals needed for the integration (call it ``K_f``).
'rslst'
    A rank-1 array of length M_f=limlst, whose first ``K_f`` elements
    contain the integral contribution over the interval
    ``(a+(k-1)c, a+kc)`` where ``c = (2*floor(|w|) + 1) * pi / |w|``
    and ``k=1,2,...,K_f``.
'erlst'
    A rank-1 array of length ``M_f`` containing the error estimate
    corresponding to the interval in the same position in
    ``infodict['rslist']``.
'ierlst'
    A rank-1 integer array of length ``M_f`` containing an error flag
    corresponding to the interval in the same position in
    ``infodict['rslist']``.  See the explanation dictionary (last entry
    in the output tuple) for the meaning of the codes.

Examples
--------
Calculate :math:`\\int^4_0 x^2 dx` and compare with an analytic result

>>> from scipy import integrate
>>> x2 = lambda x: x**2
>>> integrate.quad(x2, 0, 4)
(21.333333333333332, 2.3684757858670003e-13)
>>> print(4**3 / 3.)  # analytical result
21.3333333333

Calculate :math:`\\int^\\infty_0 e^{-x} dx`

>>> invexp = lambda x: np.exp(-x)
>>> integrate.quad(invexp, 0, np.inf)
(1.0, 5.842605999138044e-11)

>>> f = lambda x,a : a*x
>>> y, err = integrate.quad(f, 0, 1, args=(1,))
>>> y
0.5
>>> y, err = integrate.quad(f, 0, 1, args=(3,))
>>> y
1.5

Calculate :math:`\\int^1_0 x^2 + y^2 dx` with ctypes, holding
y parameter as 1::

    testlib.c =>
        double func(int n, double args[n]){
            return args[0]*args[0] + args[1]*args[1];}
    compile to library testlib.*

>>> from scipy import integrate
>>> import ctypes
>>> lib = ctypes.CDLL('/home/.../testlib.*') #use absolute path
>>> lib.func.restype = ctypes.c_double
>>> lib.func.argtypes = (ctypes.c_int,ctypes.c_double)
>>> integrate.quad(lib.func,0,1,(1))
(1.3333333333333333, 1.4802973661668752e-14)
>>> print((1.0**3/3.0 + 1.0) - (0.0**3/3.0 + 0.0)) #Analytic result
1.3333333333333333