Derivatives and integrals of hyperbolic functions
Problem 2.2448 · medium
Find \( \displaystyle \int \tanh{\left(x \right)} \, dx \). (Omit the constant of integration.)
- Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
- \[ \frac{d}{d x} \left(x - \ln{\left(\tanh{\left(x \right)} + 1 \right)}\right) = \tanh{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( x - \ln{\left(\tanh{\left(x \right)} + 1 \right)} + C \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | a central difference, with sinh and cosh computed from exponentials by mpmath |
Reviewers
gpt-oss:20b: fail (error) — The solution provides no valid integration steps; the first sentence merely lists derivative formulas and vaguely mentions a substitution, but does not actually perform or justify the integration of tanh(x).qwen3.6:27b-mlx: fail (misleading) — The solution does not actually solve the problem; it merely verifies the given answer by differentiation. The stated method (substitution) is never executed, so the student learns nothing about how to derive the integral.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution does not actually solve the problem; it merely verifies the given answer by differentiation. The stated method (substitution) is never executed, so the student learns nothing about how to derive the integral.gpt-oss:20b: fail (error) 2026-10-19 — The solution provides no valid integration steps; the first sentence merely lists derivative formulas and vaguely mentions a substitution, but does not actually perform or justify the integration of tanh(x).qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution provides no derivation, only a verification of the final answer. It fails to show the integration steps (such as substitution u = cosh(x) or splitting into sinh/cosh), which is the core of the problem.gpt-oss:20b: fail (error) 2026-10-17 — The solution provides no actual setup or justification for the integral; it merely lists derivative formulas and claims a substitution without showing how anh(x) is integrated. The first sentence does not model the problem or explain the method, so the solution is incomplete and incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution does not actually solve the problem; it merely verifies the given answer by differentiation. The first sentence is incoherent and does not describe a valid method for finding the antiderivative of tanh(x).gpt-oss:20b: fail (error) 2026-10-17 — The proposed antiderivative x−ln(tanh x+1) is incorrect; its derivative does not simplify to tanh x. The solution omits a correct substitution and fails to verify the integral properly.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/hyperbolic_calculus, checked 2026-10-19 with SymPy 1.14.0.