code: def integrate_nonlinear_no_specials(a, d, DE, z=None):
      """
      Integration of nonlinear monomials with no specials.
  
      Explanation
      ===========
  
      Given a nonlinear monomial t over k such that Sirr ({p in k[t] | p is
      special, monic, and irreducible}) is empty, and f in k(t), returns g
      elementary over k(t) and a Boolean b in {True, False} such that f - Dg is
      in k if b == True, or f - Dg does not have an elementary integral over k(t)
      if b == False.
  
      This function is applicable to all nonlinear extensions, but in the case
      where it returns b == False, it will only have proven that the integral of
      f - Dg is nonelementary if Sirr is empty.
  
      This function returns a Basic expression.
      """
      # TODO: Integral from k?
      # TODO: split out nonelementary integral
      # XXX: a and d must be canceled, or this might not return correct results
      z = z or Dummy("z")
      s = list(
          zip(reversed(DE.T), reversed([f(DE.x) for f in DE.Tfuncs]))
      )
  
      g1, h, r = hermite_reduce(a, d, DE)
      g2, b = residue_reduce(h[0], h[1], DE, z=z)
      if not b:
          return (
              (g1[0].as_expr() / g1[1].as_expr()).subs(s)
              + residue_reduce_to_basic(g2, DE, z),
              b,
          )
  
      # Because f has no specials, this should be a polynomial in t, or else
      # there is a bug.
      p = cancel(
          h[0].as_expr() / h[1].as_expr()
          - residue_reduce_derivation(g2, DE, z).as_expr()
          + r[0].as_expr() / r[1].as_expr()
      ).as_poly(DE.t)
      q1, q2 = polynomial_reduce(p, DE)
  
      if q2.expr.has(DE.t):
          b = False
      else:
          b = True
  
      ret = cancel(
          g1[0].as_expr() / g1[1].as_expr() + q1.as_expr()
      ).subs(s) + residue_reduce_to_basic(g2, DE, z)
      return (ret, b)
  
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 24 for calling a global function with example usage: integrate_nonlinear_no_specials(a, d, DE, z) and returns: ret,  b, unknown,  b