code: def ode_sol_simplicity(sol, func, trysolving=True):
      r"""
      Returns an extended integer representing how simple a solution to an ODE
      is.
  
      The following things are considered, in order from most simple to least:
  
      - ``sol`` is solved for ``func``.
      - ``sol`` is not solved for ``func``, but can be if passed to solve (e.g.,
        a solution returned by ``dsolve(ode, func, simplify=False``).
      - If ``sol`` is not solved for ``func``, then base the result on the
        length of ``sol``, as computed by ``len(str(sol))``.
      - If ``sol`` has any unevaluated :py:class:`~sympy.integrals.integrals.Integral`\s,
        this will automatically be considered less simple than any of the above.
  
      This function returns an integer such that if solution A is simpler than
      solution B by above metric, then ``ode_sol_simplicity(sola, func) <
      ode_sol_simplicity(solb, func)``.
  
      Currently, the following are the numbers returned, but if the heuristic is
      ever improved, this may change.  Only the ordering is guaranteed.
  
      +----------------------------------------------+-------------------+
      | Simplicity                                   | Return            |
      +==============================================+===================+
      | ``sol`` solved for ``func``                  | ``-2``            |
      +----------------------------------------------+-------------------+
      | ``sol`` not solved for ``func`` but can be   | ``-1``            |
      +----------------------------------------------+-------------------+
      | ``sol`` is not solved nor solvable for       | ``len(str(sol))`` |
      | ``func``                                     |                   |
      +----------------------------------------------+-------------------+
      | ``sol`` contains an                          | ``oo``            |
      | :obj:`~sympy.integrals.integrals.Integral`   |                   |
      +----------------------------------------------+-------------------+
  
      ``oo`` here means the SymPy infinity, which should compare greater than
      any integer.
  
      If you already know :py:meth:`~sympy.solvers.solvers.solve` cannot solve
      ``sol``, you can use ``trysolving=False`` to skip that step, which is the
      only potentially slow step.  For example,
      :py:meth:`~sympy.solvers.ode.dsolve` with the ``simplify=False`` flag
      should do this.
  
      If ``sol`` is a list of solutions, if the worst solution in the list
      returns ``oo`` it returns that, otherwise it returns ``len(str(sol))``,
      that is, the length of the string representation of the whole list.
  
      Examples
      ========
  
      This function is designed to be passed to ``min`` as the key argument,
      such as ``min(listofsolutions, key=lambda i: ode_sol_simplicity(i,
      f(x)))``.
  
      >>> from sympy import symbols, Function, Eq, tan, Integral
      >>> from sympy.solvers.ode.ode import ode_sol_simplicity
      >>> x, C1, C2 = symbols('x, C1, C2')
      >>> f = Function('f')
  
      >>> ode_sol_simplicity(Eq(f(x), C1*x**2), f(x))
      -2
      >>> ode_sol_simplicity(Eq(x**2 + f(x), C1), f(x))
      -1
      >>> ode_sol_simplicity(Eq(f(x), C1*Integral(2*x, x)), f(x))
      oo
      >>> eq1 = Eq(f(x)/tan(f(x)/(2*x)), C1)
      >>> eq2 = Eq(f(x)/tan(f(x)/(2*x) + f(x)), C2)
      >>> [ode_sol_simplicity(eq, f(x)) for eq in [eq1, eq2]]
      [28, 35]
      >>> min([eq1, eq2], key=lambda i: ode_sol_simplicity(i, f(x)))
      Eq(f(x)/tan(f(x)/(2*x)), C1)
  
      """
      # TODO: if two solutions are solved for f(x), we still want to be
      # able to get the simpler of the two
  
      # See the docstring for the coercion rules.  We check easier (faster)
      # things here first, to save time.
  
      if iterable(sol):
          # See if there are Integrals
          for i in sol:
              if (
                  ode_sol_simplicity(i, func, trysolving=trysolving)
                  == oo
              ):
                  return oo
  
          return len(str(sol))
  
      if sol.has(Integral):
          return oo
  
      # Next, try to solve for func.  This code will change slightly when CRootOf
      # is implemented in solve().  Probably a CRootOf solution should fall
      # somewhere between a normal solution and an unsolvable expression.
  
      # First, see if they are already solved
      if (
          sol.lhs == func
          and not sol.rhs.has(func)
          or sol.rhs == func
          and not sol.lhs.has(func)
      ):
          return -2
      # We are not so lucky, try solving manually
      if trysolving:
          try:
              sols = solve(sol, func)
              if not sols:
                  raise NotImplementedError
          except NotImplementedError:
              pass
          else:
              return -1
  
      # Finally, a naive computation based on the length of the string version
      # of the expression.  This may favor combined fractions because they
      # will not have duplicate denominators, and may slightly favor expressions
      # with fewer additions and subtractions, as those are separated by spaces
      # by the printer.
  
      # Additional ideas for simplicity heuristics are welcome, like maybe
      # checking if a equation has a larger domain, or if constantsimp has
      # introduced arbitrary constants numbered higher than the order of a
      # given ODE that sol is a solution of.
      return len(str(sol))
  
imports: from sympy.core import Add, S, Mul, Pow, oo
  from sympy.core.containers import Tuple
  from sympy.core.expr import AtomicExpr, Expr
  from sympy.core.function import Function, Derivative, AppliedUndef, diff, expand, expand_mul, Subs
  from sympy.core.multidimensional import vectorize
  from sympy.core.numbers import nan, zoo, Number
  from sympy.core.relational import Equality, Eq
  from sympy.core.sorting import default_sort_key, ordered
  from sympy.core.symbol import Symbol, Wild, Dummy, symbols
  from sympy.core.sympify import sympify
  from sympy.core.traversal import preorder_traversal
  from sympy.logic.boolalg import BooleanAtom, BooleanTrue, BooleanFalse
  from sympy.functions import exp, log, sqrt
  from sympy.functions.combinatorial.factorials import factorial
  from sympy.integrals.integrals import Integral
  from sympy.polys import Poly, terms_gcd, PolynomialError, lcm
  from sympy.polys.polytools import cancel
  from sympy.series import Order
  from sympy.series.series import series
  from sympy.simplify import collect, logcombine, powsimp, separatevars, simplify, cse
  from sympy.simplify.radsimp import collect_const
  from sympy.solvers import checksol, solve
  from sympy.utilities import numbered_symbols
  from sympy.utilities.iterables import uniq, sift, iterable
  from sympy.solvers.deutils import _preprocess, ode_order, _desolve
  from .single import SingleODEProblem, SingleODESolver, solver_map
label_desc: Use image node_id 14 for calling a global function with example usage: ode_sol_simplicity(sol, func, trysolving) and returns: len, len, oo, unknown, unknown, oo