code: def risch_integrate(
      f,
      x,
      extension=None,
      handle_first="log",
      separate_integral=False,
      rewrite_complex=None,
      conds="piecewise",
  ):
      r"""
      The Risch Integration Algorithm.
  
      Explanation
      ===========
  
      Only transcendental functions are supported.  Currently, only exponentials
      and logarithms are supported, but support for trigonometric functions is
      forthcoming.
  
      If this function returns an unevaluated Integral in the result, it means
      that it has proven that integral to be nonelementary.  Any errors will
      result in raising NotImplementedError.  The unevaluated Integral will be
      an instance of NonElementaryIntegral, a subclass of Integral.
  
      handle_first may be either 'exp' or 'log'.  This changes the order in
      which the extension is built, and may result in a different (but
      equivalent) solution (for an example of this, see issue 5109).  It is also
      possible that the integral may be computed with one but not the other,
      because not all cases have been implemented yet.  It defaults to 'log' so
      that the outer extension is exponential when possible, because more of the
      exponential case has been implemented.
  
      If ``separate_integral`` is ``True``, the result is returned as a tuple (ans, i),
      where the integral is ans + i, ans is elementary, and i is either a
      NonElementaryIntegral or 0.  This useful if you want to try further
      integrating the NonElementaryIntegral part using other algorithms to
      possibly get a solution in terms of special functions.  It is False by
      default.
  
      Examples
      ========
  
      >>> from sympy.integrals.risch import risch_integrate
      >>> from sympy import exp, log, pprint
      >>> from sympy.abc import x
  
      First, we try integrating exp(-x**2). Except for a constant factor of
      2/sqrt(pi), this is the famous error function.
  
      >>> pprint(risch_integrate(exp(-x**2), x))
        /
       |
       |    2
       |  -x
       | e    dx
       |
      /
  
      The unevaluated Integral in the result means that risch_integrate() has
      proven that exp(-x**2) does not have an elementary anti-derivative.
  
      In many cases, risch_integrate() can split out the elementary
      anti-derivative part from the nonelementary anti-derivative part.
      For example,
  
      >>> pprint(risch_integrate((2*log(x)**2 - log(x) - x**2)/(log(x)**3 -
      ... x**2*log(x)), x))
                                               /
                                              |
        log(-x + log(x))   log(x + log(x))    |   1
      - ---------------- + --------------- +  | ------ dx
               2                  2           | log(x)
                                              |
                                             /
  
      This means that it has proven that the integral of 1/log(x) is
      nonelementary.  This function is also known as the logarithmic integral,
      and is often denoted as Li(x).
  
      risch_integrate() currently only accepts purely transcendental functions
      with exponentials and logarithms, though note that this can include
      nested exponentials and logarithms, as well as exponentials with bases
      other than E.
  
      >>> pprint(risch_integrate(exp(x)*exp(exp(x)), x))
       / x\
       \e /
      e
      >>> pprint(risch_integrate(exp(exp(x)), x))
        /
       |
       |  / x\
       |  \e /
       | e     dx
       |
      /
  
      >>> pprint(risch_integrate(x*x**x*log(x) + x**x + x*x**x, x))
         x
      x*x
      >>> pprint(risch_integrate(x**x, x))
        /
       |
       |  x
       | x  dx
       |
      /
  
      >>> pprint(risch_integrate(-1/(x*log(x)*log(log(x))**2), x))
           1
      -----------
      log(log(x))
  
      """
      f = S(f)
  
      DE = extension or DifferentialExtension(
          f,
          x,
          handle_first=handle_first,
          dummy=True,
          rewrite_complex=rewrite_complex,
      )
      fa, fd = DE.fa, DE.fd
  
      result = S.Zero
      for case in reversed(DE.cases):
          if (
              not fa.expr.has(DE.t)
              and not fd.expr.has(DE.t)
              and not case == "base"
          ):
              DE.decrement_level()
              fa, fd = frac_in((fa, fd), DE.t)
              continue
  
          fa, fd = fa.cancel(fd, include=True)
          if case == "exp":
              ans, i, b = integrate_hyperexponential(
                  fa, fd, DE, conds=conds
              )
          elif case == "primitive":
              ans, i, b = integrate_primitive(fa, fd, DE)
          elif case == "base":
              # XXX: We can't call ratint() directly here because it doesn't
              # handle polynomials correctly.
              ans = integrate(
                  fa.as_expr() / fd.as_expr(), DE.x, risch=False
              )
              b = False
              i = S.Zero
          else:
              raise NotImplementedError(
                  "Only exponential and logarithmic "
                  "extensions are currently supported."
              )
  
          result += ans
          if b:
              DE.decrement_level()
              fa, fd = frac_in(i, DE.t)
          else:
              result = result.subs(DE.backsubs)
              if not i.is_zero:
                  i = NonElementaryIntegral(
                      i.function.subs(DE.backsubs), i.limits
                  )
              if not separate_integral:
                  result += i
                  return result
              else:
                  if isinstance(i, NonElementaryIntegral):
                      return (result, i)
                  else:
                      return (result, 0)
  
imports: from types import GeneratorType
  from functools import reduce
  from sympy.core.function import Lambda
  from sympy.core.mul import Mul
  from sympy.core.intfunc import ilcm
  from sympy.core.numbers import I, oo
  from sympy.core.power import Pow
  from sympy.core.relational import Ne
  from sympy.core.singleton import S
  from sympy.core.sorting import ordered, default_sort_key
  from sympy.core.symbol import Dummy, Symbol
  from sympy.functions.elementary.exponential import log, exp
  from sympy.functions.elementary.hyperbolic import cosh, coth, sinh, tanh
  from sympy.functions.elementary.piecewise import Piecewise
  from sympy.functions.elementary.trigonometric import atan, sin, cos, tan, acot, cot, asin, acos
  from .integrals import integrate, Integral
  from .heurisch import _symbols
  from sympy.polys.polyerrors import DomainError, PolynomialError
  from sympy.polys.polytools import real_roots, cancel, Poly, gcd, reduced
  from sympy.polys.rootoftools import RootSum
  from sympy.utilities.iterables import numbered_symbols
label_desc: Use image node_id 25 for calling a global function with example usage: risch_integrate(f, x, extension, handle_first, separate_integral, rewrite_complex, conds) and returns: result, result,  i, result,  int