code: def _osbornei(e, d):
      """Replace all trig functions with hyperbolic functions using
      the Osborne rule.
  
      Notes
      =====
  
      ``d`` is a dummy variable to prevent automatic evaluation
      of trigonometric/hyperbolic functions.
  
      References
      ==========
  
      .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
      """
  
      def f(rv):
          if not isinstance(rv, TrigonometricFunction):
              return rv
          const, x = rv.args[0].as_independent(d, as_Add=True)
          a = x.xreplace({d: S.One}) + const * I
          if isinstance(rv, sin):
              return sinh(a) / I
          elif isinstance(rv, cos):
              return cosh(a)
          elif isinstance(rv, tan):
              return tanh(a) / I
          elif isinstance(rv, cot):
              return coth(a) * I
          elif isinstance(rv, sec):
              return sech(a)
          elif isinstance(rv, csc):
              return csch(a) * I
          else:
              raise NotImplementedError("unhandled %s" % rv.func)
  
      return bottom_up(e, f)
  
imports: from collections import defaultdict
  from sympy.core.add import Add
  from sympy.core.expr import Expr
  from sympy.core.exprtools import Factors, gcd_terms, factor_terms
  from sympy.core.function import expand_mul
  from sympy.core.mul import Mul
  from sympy.core.numbers import pi, I
  from sympy.core.power import Pow
  from sympy.core.singleton import S
  from sympy.core.sorting import ordered
  from sympy.core.symbol import Dummy
  from sympy.core.sympify import sympify
  from sympy.core.traversal import bottom_up
  from sympy.functions.combinatorial.factorials import binomial
  from sympy.functions.elementary.hyperbolic import cosh, sinh, tanh, coth, sech, csch, HyperbolicFunction
  from sympy.functions.elementary.trigonometric import cos, sin, tan, cot, sec, csc, sqrt, TrigonometricFunction
  from sympy.ntheory.factor_ import perfect_power
  from sympy.polys.polytools import factor
  from sympy.strategies.tree import greedy
  from sympy.strategies.core import identity, debug
  from sympy import SYMPY_DEBUG
label_desc: Use image node_id 34 for calling a global function with example usage: _osbornei(e, d) and returns: bottom_up, rv, unknown, cosh, unknown, unknown, sech, unknown