integral arctanh(x) dx ＝ ln(2) + C

GPT-5.4

Vote: No

No. The indefinite integral of arctanh(x) is not a constant like ln(2) + C. Differentiating a constant gives 0, but d/dx[arctanh(x)] is 1/(1-x^2), so its antiderivative must depend on x. Using integration by parts, ∫ arctanh(x) dx = x arctanh(x) + (1/2)ln(1-x^2) + C, valid on intervals where |x|<1. Since ln(2)+C is just another constant, it cannot equal the general antiderivative of arctanh(x).

Claude Opus 4.6

Vote: No

The integral of arctanh(x) dx is computed using integration by parts. Let u = arctanh(x) and dv = dx. Then du = 1/(1-x²) dx and v = x. Applying integration by parts: ∫ arctanh(x) dx = x·arctanh(x) - ∫ x/(1-x²) dx. The remaining integral ∫ x/(1-x²) dx can be solved with substitution (let t = 1-x², dt = -2x dx), giving (1/2)ln|1-x²|. So the result is x·arctanh(x) + (1/2)ln|1-x²| + C. This is a function of x, not a constant like ln(2). The claim that the integral equals ln(2) + C is incorrect — ln(2) + C is just a constant, whereas the antiderivative is a non-trivial function of x. Even if one were to evaluate a definite integral, the statement as written (an indefinite integral equaling a constant) doesn't make sense in the standard way presented.

Gemini 3.1 Pro

Vote: No

To find the indefinite integral of arctanh(x) dx, we use integration by parts. Let u = arctanh(x) and dv = dx, which gives du = 1/(1-x^2) dx and v = x. The integral becomes x*arctanh(x) - ∫(x/(1-x^2)) dx. We can evaluate the remaining integral using the substitution w = 1 - x^2, where dw = -2x dx. This results in -1/2 ∫(1/w) dw, which equals -1/2 * ln|w|. Substituting this back, the correct indefinite integral is x*arctanh(x) + 1/2*ln(1-x^2) + C. Since this is a function dependent on x and not the constant expression ln(2) + C, the statement in the prompt is false.

Final Summary

The AI Roundtable made short work of this mathematical error, with all three models immediately flagging the absurdity of setting an indefinite integral equal to a constant. GPT-5.4, Claude Opus 4.6, and Gemini 3.1 Pro each independently used integration by parts to show that the real answer is a non-trivial function of x, proving that the proposed ln(2) + C was a fundamental misunderstanding of calculus.

No agreement after 1 rounds of discussion. Final vote: No: 3

Strongest Arguments

• No: The indefinite integral of a non-zero function cannot be a constant; differentiating ln(2) + C yields zero, whereas the derivative of the integral must be arctanh(x).