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Write a quotient by a radical sum with a constant conjugate product over the conjugate - #1962
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Rafael-SOWNet merged 1 commit intoOct 20, 2026
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…ver the conjugate e^asech(a x) arrives as 1/(a x) + sqrt(1/(a x)^2 - 1), whose product with its conjugate is 1, so x^m/e^asech(a x) is x^m (1/(a x) - sqrt(1/(a x)^2 - 1)), a sum with no quotient left. (A + B sqrt(R))(A - B sqrt(R)) = A^2 - B^2 R holds on every branch, so the rewrite holds wherever the quotient is defined. Rubi's 7.5.2 x^4/e^asech(a x) to 1/(e^asech(a x) x) were declined. The search took the quotient as written, a root below the bar beside a pole. The rule is asked only of the question asked. By parts against asech or acsch asks a quotient by the same sum, and rewritten over the conjugate those sub-questions ran past the budget where the rules below answer them as written. Asked below the top, the rule cost x*asech(a*x)^2 and five of its neighbours, which master answers in 130-230 ms. 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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e^asech(a x)arrives as1/(a x) + sqrt(1/(a x)^2 - 1), whose product with its conjugate is 1. Sox^m/e^asech(a x)isx^m (1/(a x) - sqrt(1/(a x)^2 - 1)), a sum with no quotient left. The identity(A + B sqrt(R))(A - B sqrt(R)) = A^2 - B^2 Rholds on every branch, so the rewrite holds wherever the quotient is defined. Rubi's 7.5.2x^4/e^asech(a x)to1/(e^asech(a x) x)were declined. The answers hold where|a x| <= 1, which is where the integrand is real.The rule fires only on the question asked. By parts against
asechoracschasks a quotient by the same sum. A first version that also fired on those sub-questions lostx*asech(a*x)^2and five similar problems, which master answers in 130–230 ms. They are now in the test.Measured on
8a1ef454, branch against master:Re-run alone on both builds, 7.5.2:111–114 and 116 go from timed out or declined to answered, in 30 ms to 1.1 s. 6.4.2:50 and 7.4.1:369 both take about 24 s on either build, so they flip with load: one was counted solved on master only, the other on the branch only. The other moves time out on both builds. No run had a wrong answer.
On this master the dispatch line sits after #1960's, the last rule of its group.
AQuotientByARadicalSumWithAConstantConjugateProductIntegralTestdifferentiates the answers back and checks the six problems by parts stay answered.BREAKING-CHANGES.mdhas the entry, with the 2.5.0 column measured on v2.5.0.Part of #233.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura