code: def limited_integrate_reduce(fa, fd, G, DE):
      """
      Simpler version of step 1 & 2 for the limited integration problem.
  
      Explanation
      ===========
  
      Given a derivation D on k(t) and f, g1, ..., gn in k(t), return
      (a, b, h, N, g, V) such that a, b, h in k[t], N is a non-negative integer,
      g in k(t), V == [v1, ..., vm] in k(t)^m, and for any solution v in k(t),
      c1, ..., cm in C of f == Dv + Sum(ci*wi, (i, 1, m)), p = v*h is in k<t>, and
      p and the ci satisfy a*Dp + b*p == g + Sum(ci*vi, (i, 1, m)).  Furthermore,
      if S1irr == Sirr, then p is in k[t], and if t is nonlinear or Liouvillian
      over k, then deg(p) <= N.
  
      So that the special part is always computed, this function calls the more
      general prde_special_denom() automatically if it cannot determine that
      S1irr == Sirr.  Furthermore, it will automatically call bound_degree() when
      t is linear and non-Liouvillian, which for the transcendental case, implies
      that Dt == a*t + b with for some a, b in k*.
      """
      dn, ds = splitfactor(fd, DE)
      E = [splitfactor(gd, DE) for _, gd in G]
      En, Es = list(zip(*E))
      c = reduce(
          lambda i, j: i.lcm(j), (dn,) + En
      )  # lcm(dn, en1, ..., enm)
      hn = c.gcd(c.diff(DE.t))
      a = hn
      b = -derivation(hn, DE)
      N = 0
  
      # These are the cases where we know that S1irr = Sirr, but there could be
      # others, and this algorithm will need to be extended to handle them.
      if DE.case in ("base", "primitive", "exp", "tan"):
          hs = reduce(
              lambda i, j: i.lcm(j), (ds,) + Es
          )  # lcm(ds, es1, ..., esm)
          a = hn * hs
          b -= (hn * derivation(hs, DE)).quo(hs)
          mu = min(
              order_at_oo(fa, fd, DE.t),
              min([order_at_oo(ga, gd, DE.t) for ga, gd in G]),
          )
          # So far, all the above are also nonlinear or Liouvillian, but if this
          # changes, then this will need to be updated to call bound_degree()
          # as per the docstring of this function (DE.case == 'other_linear').
          N = (
              hn.degree(DE.t)
              + hs.degree(DE.t)
              + max(0, 1 - DE.d.degree(DE.t) - mu)
          )
      else:
          # TODO: implement this
          raise NotImplementedError
  
      V = [(-a * hn * ga).cancel(gd, include=True) for ga, gd in G]
      return (a, b, a, N, (a * hn * fa).cancel(fd, include=True), V)
  
imports: import itertools
  from functools import reduce
  from sympy.core.intfunc import ilcm
  from sympy.core import Dummy, Add, Mul, Pow, S
  from sympy.integrals.rde import order_at, order_at_oo, weak_normalizer, bound_degree
  from sympy.integrals.risch import gcdex_diophantine, frac_in, derivation, residue_reduce, splitfactor, residue_reduce_derivation, DecrementLevel, recognize_log_derivative
  from sympy.polys import Poly, lcm, cancel, sqf_list
  from sympy.polys.polymatrix import PolyMatrix
  from sympy.solvers import solve
label_desc: Use image node_id 13 for calling a global function with example usage: limited_integrate_reduce(fa, fd, G, DE) and returns: a,  b,  a,  N,  unknown,  V