← suite simplipy

sp-acj4-196-cond2 true

category simplipy-conditional

Summary

Given
\(x \in \mathbb{C},\quad x \in i\mathbb{R},\quad \cosh x \neq 0,\quad \sin\left(\tanh x\right) \neq 1,\quad \sin\left(\tanh x\right) \neq -1,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\sin\left(\tanh x\right)\right) = \operatorname{arsinh}\left(\tan\left(\tanh x\right)\right)\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0 -> asinh tan tanh _0 [weakest assumption found: imaginary]

Lean proof: included

Answers (latest run per system; open code for the reproducer that ran)

systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
none
always-noneno verdictnull0.02026-10-13 17:21:18 ba9dc4def822(imaginary x) ∧ (nonzero (cosh x)) ∧ (!= (sin (tanh x)) 1) ∧ (!= (sin (tanh x)) -1) ∧ (nonzero (cos (tanh x))) ⊢ (= (atanh (sin (tanh x))) (asinh (tan (tanh x))))null
sympy
sympy-askno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · ask → None0.1372026-10-13 17:21:19 ba9dc4def822Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cosh(x)), Ne(sin(tanh(x)), 1, evaluate=False), Ne(sin(tanh(x)), -1, evaluate=False), ~Q.zero(cos(tanh(x))))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · ask → None0.042026-10-12 21:54:02 ee702ae2fe06Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cosh(x)), Ne(sin(tanh(x)), 1, evaluate=False), Ne(sin(tanh(x)), -1, evaluate=False), ~Q.zero(cos(tanh(x))))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · refine → -asinh(tan(tanh(x))) + atanh(sin(tanh(x)))0.0442026-10-13 17:22:30 ba9dc4def822Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cosh(x)), Ne(sin(tanh(x)), 1, evaluate=False), Ne(sin(tanh(x)), -1, evaluate=False), ~Q.zero(cos(tanh(x))))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
refinerefine(Residual, Assum)→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
answernull
sympy-refineno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · refine → -asinh(tan(tanh(x))) + atanh(sin(tanh(x)))0.062026-10-12 21:54:16 ee702ae2fe06Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cosh(x)), Ne(sin(tanh(x)), 1, evaluate=False), Ne(sin(tanh(x)), -1, evaluate=False), ~Q.zero(cos(tanh(x))))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
refinerefine(Residual, Assum)→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
answernull
sympy-simplifyno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · simplify → -asinh(tan(tanh(x))) + atanh(sin(tanh(x)))0.0762026-10-13 17:24:01 ba9dc4def822Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
# dropped (not a unary fact about one symbol): Ne(sin(tanh(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sin(tanh(x)), -1, evaluate=False)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(tanh(x)))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
simplifysimplify(Residual)→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
answernull
sympy-simplifyno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · simplify → -asinh(tan(tanh(x))) + atanh(sin(tanh(x)))0.0812026-10-12 21:54:43 ee702ae2fe06Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
# dropped (not a unary fact about one symbol): Ne(sin(tanh(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sin(tanh(x)), -1, evaluate=False)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(tanh(x)))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
simplifysimplify(Residual)→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
answernull
wolfram
wolfram-fullsimplify no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]2.1372026-10-13 17:24:42 ba9dc4def822Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultFullSimplify[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
wolfram-refine no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]0.0222026-10-13 17:24:31 ba9dc4def822Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultRefine[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
wolfram-simplify no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]0.1032026-10-13 17:24:17 ba9dc4def822Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultSimplify[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]2.1252026-10-13 16:43:00 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultFullSimplify[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
wolfram-refine no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]0.0232026-10-13 16:42:50 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultRefine[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
wolfram-simplify no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]0.12026-10-13 16:42:35 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultSimplify[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
sympy-simplifyno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · simplify → -asinh(tan(tanh(x))) + atanh(sin(tanh(x)))0.0782026-10-13 16:42:18 3ba0dfac4bcaQ.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
# dropped (not a unary fact about one symbol): Ne(sin(tanh(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sin(tanh(x)), -1, evaluate=False)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(tanh(x)))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
simplifysimplify(Residual)→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
answernull
sympy-refineno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · refine → -asinh(tan(tanh(x))) + atanh(sin(tanh(x)))0.0442026-10-13 16:40:40 3ba0dfac4bcaQ.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cosh(x)), Ne(sin(tanh(x)), 1, evaluate=False), Ne(sin(tanh(x)), -1, evaluate=False), ~Q.zero(cos(tanh(x))))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
refinerefine(Residual, Assum)→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(imaginary x) ∧ (nonzero (cosh x)) ∧ (!= (sin (tanh x)) 1) ∧ (!= (sin (tanh x)) -1) ∧ (nonzero (cos (tanh x))) ⊢ (= (atanh (sin (tanh x))) (asinh (tan (tanh x))))null
sympy-askno verdictnulldifference → -asinh(tan(tanh(x))) + atanh(sin(tanh(x))) · ask → None0.1362026-10-13 16:39:28 3ba0dfac4bcaQ.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ∧ Ne(sin(tanh(x)), 1) ∧ Ne(sin(tanh(x)), (-1)) ∧ (~Q.zero(cos(tanh(x)))) ⊢ Eq(atanh(sin(tanh(x))), asinh(tan(tanh(x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), ~Q.zero(cosh(x)), Ne(sin(tanh(x)), 1, evaluate=False), Ne(sin(tanh(x)), -1, evaluate=False), ~Q.zero(cos(tanh(x))))
Residual = atanh(sin(tanh(x))) - asinh(tan(tanh(x)))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differenceatanh(sin(tanh(x))) - asinh(tan(tanh(x)))→-asinh(tan(tanh(x))) + atanh(sin(tanh(x)))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]2.132026-10-12 21:55:24 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultFullSimplify[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
wolfram-refine no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]0.0232026-10-12 21:55:14 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultRefine[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(imaginary x) ∧ (nonzero (cosh x)) ∧ (!= (sin (tanh x)) 1) ∧ (!= (sin (tanh x)) -1) ∧ (nonzero (cos (tanh x))) ⊢ (= (atanh (sin (tanh x))) (asinh (tan (tanh x))))null
wolfram-simplify no verdictnulldifference → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]] · result → -ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]0.1012026-10-12 21:55:00 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ∧ (Sin[Tanh[x]] != 1) ∧ (Sin[Tanh[x]] != (-1)) ∧ (Cos[Tanh[x]] != 0) ⊢ (ArcTanh[Sin[Tanh[x]]] == ArcSinh[Tan[Tanh[x]]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0)) && ((Sin[Tanh[x]] != 1)) && ((Sin[Tanh[x]] != (-1))) && ((Cos[Tanh[x]] != 0));
  Residual = (ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Sin[Tanh[x]]]) - (ArcSinh[Tan[Tanh[x]]])→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
resultSimplify[Residual, Assum]→-ArcSinh[Tan[Tanh[x]]] + ArcTanh[Sin[Tanh[x]]]
answernull

Source (conditional.sexp)

(case sp-acj4-196-cond2 (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0  ->  asinh tan tanh _0  [weakest assumption found: imaginary]")
  (var x complex)
  (assume (imaginary x))
  (assume (nonzero (cosh x)))
  (assume (!= (sin (tanh x)) 1))
  (assume (!= (sin (tanh x)) -1))
  (assume (nonzero (cos (tanh x))))
  (ask (= (atanh (sin (tanh x))) (asinh (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨y, hy, rfl⟩ := AE.cond_imaginary_exists h0.2; have hS : AE.tanh (↑y * Complex.I) = ↑(Real.tan y) * Complex.I := (by rw [AE.tanh_eq, AE.cond_tanh_mul_I']); rw [hS] at h2 h3 h4 ⊢; rw [AE.sin_eq, AE.cond_sin_mul_I'] at h2 h3 ⊢; refine ⟨⟨⟨⟨h2.1.1, by simp⟩, (AE.atanh.D_iff _).mpr ⟨h2.2, h3.2⟩⟩, ⟨⟨h2.1.1, (AE.tan.D_iff _).mpr (by simpa using h4.2)⟩, by simp⟩⟩, ?_⟩; rw [AE.cond_atanh_mul_I, AE.tan_eq, AE.cond_tan_mul_I', AE.cond_asinh_mul_I_of (AE.cond_tanh_mem _).1 (AE.cond_tanh_mem _).2, AE.cond_arctan_sinh]"))