class_docstr: The Generalized Regularized Incomplete Beta function is given by
  
  .. math::
      \mathrm{I}_{(x_1, x_2)}(a, b) = \frac{\mathrm{B}_{(x_1, x_2)}(a, b)}{\mathrm{B}(a, b)}
  
  The Regularized Incomplete Beta function is a special case
  of the Generalized Regularized Incomplete Beta function :
  
  .. math:: \mathrm{I}_z (a, b) = \mathrm{I}_{(0, z)}(a, b)
  
  The Regularized Incomplete Beta function is the cumulative distribution
  function of the beta distribution.
  
  Examples
  ========
  
  >>> from sympy import betainc_regularized, symbols, conjugate
  >>> a, b, x, x1, x2 = symbols('a b x x1 x2')
  
  The Generalized Regularized Incomplete Beta
  function is given by:
  
  >>> betainc_regularized(a, b, x1, x2)
  betainc_regularized(a, b, x1, x2)
  
  The Regularized Incomplete Beta function
  can be obtained as follows:
  
  >>> betainc_regularized(a, b, 0, x)
  betainc_regularized(a, b, 0, x)
  
  The Regularized Incomplete Beta function
  obeys the mirror symmetry:
  
  >>> conjugate(betainc_regularized(a, b, x1, x2))
  betainc_regularized(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))
  
  We can numerically evaluate the Regularized Incomplete Beta function
  to arbitrary precision for any complex numbers a, b, x1 and x2:
  
  >>> from sympy import betainc_regularized, pi, E
  >>> betainc_regularized(1, 2, 0, 0.25).evalf(10)
  0.4375000000
  >>> betainc_regularized(pi, E, 0, 1).evalf(5)
  1.00000
  
  The Generalized Regularized Incomplete Beta function can be
  expressed in terms of the Generalized Hypergeometric function.
  
  >>> from sympy import hyper
  >>> betainc_regularized(a, b, x1, x2).rewrite(hyper)
  (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/(a*beta(a, b))
  
  See Also
  ========
  
  beta: Beta function
  hyper: Generalized Hypergeometric function
  
  References
  ==========
  
  .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
  .. [2] https://dlmf.nist.gov/8.17
  .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
  .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/
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code: def __new__(cls, a, b, x1, x2):
      return Function.__new__(cls, a, b, x1, x2)
  
imports: from sympy.core import S
  from sympy.core.function import Function, ArgumentIndexError
  from sympy.core.symbol import Dummy
  from sympy.functions.special.gamma_functions import gamma, digamma
  from sympy.functions.combinatorial.numbers import catalan
  from sympy.functions.elementary.complexes import conjugate
label_desc: Use image node_id 1 for calling the betainc_regularized obj's underlying member method code with example usage: obj.__new__(cls, a, b, x1, x2) and returns: Function