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Integrate an inverse hyperbolic function by parts against its logarithm - #1969
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Rafael-SOWNet merged 1 commit intoOct 20, 2026
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The parser writes each inverse hyperbolic function as a logarithm, ln(A + sqrt(A^2 + k)) or ln((1 + u)/(1 - u)). Differentiated as written, that is a quotient with the root in both parts, which the search never brought down to the root alone. Its derivative is now written as the algebraic function it is, A'/sqrt(A^2 + k) or 2 u'/(1 - u^2), and P + Q L is integrated as integral P + G L - integral G L', with G the antiderivative of Q. Rubi's x (a + b acosh(c x))/(d + e x^2)^2, x^3 (a + b asech(c x))/(d + e x^2)^3, asech(a + b x)/x^3 and acsch(a + b x)/x^3 were past the budget. The rule declines where: - Q is a polynomial, which is answered already, or not rational: beside a root, the rules for the inverse function answer, and the root's integral asked here took their time; - G holds L itself; - the answer holds on part of the line only; - a factor below Q's bar divides the function's own quadratic, as in atanh(a x)/(1 - a^2 x^2)^3, which the substitution under the function answers; - an artanh's argument is not linear. Part of #233. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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October 29, 2026 00:43
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The parser writes each inverse hyperbolic function as a logarithm,
ln(A + sqrt(A^2 + k))orln((1 + u)/(1 - u)). Differentiated as written, that is a quotient with the root in both parts, which the search never brought down to the root alone. The rule writes the derivative as the algebraic function it is,A'/sqrt(A^2 + k)or2 u'/(1 - u^2), and integratesP + Q Las∫P + G L - ∫G L', withGthe antiderivative ofQ.It declines where:
Qis a polynomial, which is already answered;Qis not rational: beside a root, the rules for the inverse function answer, and the root's integral asked here first took their time;GholdsLitself;Q's bar divides the function's own quadratic, as inatanh(a x)/(1 - a^2 x^2)^3, which the substitution under the function answers;Each limit came from a measured loss. Validated by chain four times, on
8a1ef454, branch against master:Qrational: one wrong answer, the master defect fixed in Take an interval's size without squaring, so a size past the doubles does not agree #1966, and two losses.Re-run alone on both builds, 10 problems go from timed out to answered: Rubi's 7.1.4:20, 7.2.4:621/629/630, 7.5.1:165, 7.5.2:14/15, 7.6.1:151/152 and 7.6.2:13. None of master's answers is lost, and no run had a wrong answer.
In that chain, five 7.3.7 problems that master declines in 1–11 s timed out instead, all
atanh(x sqrt(e)/sqrt(d + e x^2))/x^n. The last limit, a linear argument for the artanh, was added for them after the chain. Probed on this rebased branch, those problems answer, through #1952's rule, and so do all ten gains and both earlier losses: 13 of 13.A sweep of the 55 rows with one linear inverse hyperbolic function and a whole power of a quadratic below the bar: master answers 49, the branch 55.
ByPartsAgainstAnInverseHyperbolicFunctionAsItsLogarithmIntegralTestincludes 7.6.1:130 and both losses.BREAKING-CHANGES.mdhas the entry.Part of #233.
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