code: def test_construct_d_case_6():
      """
      This function tests the Case 6 in the step
      to calculate coefficients of the d-vector.
  
      Each test case has 3 values -
  
      1. num - Numerator of the rational function a(x).
      2. den - Denominator of the rational function a(x).
      3. d - The d-vector.
      """
      tests = [
          (
              Poly(-2 * x**2 - 5, x, domain="ZZ"),
              Poly(
                  4 * x**4 + 2 * x**2 + 10 * x + 2, x, domain="ZZ"
              ),
              [[S(1) / 2 + I / 2], [S(1) / 2 - I / 2]],
          ),
          (
              Poly(
                  -2 * x**3 - 4 * x**2 - 2 * x - 5, x, domain="ZZ"
              ),
              Poly(
                  x**6
                  - x**5
                  + 2 * x**4
                  - 4 * x**3
                  - 5 * x**2
                  - 5 * x
                  + 9,
                  x,
                  domain="ZZ",
              ),
              [[1], [0]],
          ),
          (
              Poly(-5 * x**3 + x**2 + 11 * x + 12, x, domain="ZZ"),
              Poly(
                  6 * x**8
                  - 26 * x**7
                  - 27 * x**6
                  - 10 * x**5
                  - 44 * x**4
                  - 46 * x**3
                  - 34 * x**2
                  - 27 * x
                  - 42,
                  x,
                  domain="ZZ",
              ),
              [[1], [0]],
          ),
      ]
      for num, den, d in tests:
          assert construct_d_case_6(num, den, x) == d
  
imports: from sympy.core.random import randint
  from sympy.core.function import Function
  from sympy.core.mul import Mul
  from sympy.core.numbers import I, Rational, oo
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy, symbols
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.hyperbolic import tanh
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.trigonometric import sin
  from sympy.polys.polytools import Poly
  from sympy.simplify.ratsimp import ratsimp
  from sympy.solvers.ode.subscheck import checkodesol
  from sympy.testing.pytest import slow
  from sympy.solvers.ode.riccati import riccati_normal, riccati_inverse_normal, riccati_reduced, match_riccati, inverse_transform_poly, limit_at_inf, check_necessary_conds, val_at_inf, construct_c_case_1, construct_c_case_2, construct_c_case_3, construct_d_case_4, construct_d_case_5, construct_d_case_6, rational_laurent_series, solve_riccati
label_desc: Use image node_id 17 for calling a global function with example usage: test_construct_d_case_6() without return types