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Integrate half-odd powers of a ± a sin beside a fractional power of the cosine - #1964
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Rafael-SOWNet merged 1 commit intoOct 20, 2026
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…he cosine Rubi's (e cos(c + d x))^p (a + a sin(c + d x))^m with half-odd p and m were past the budget: the substitution search spent it on the two radicals. Under u = sin(y) they are (1 + u)^A (1 - u)^B with A and B quarter-odd, which by Chebyshev's theorem is elementary only where A + B = m + p - 1 is whole. With every power a number, the rule now asks only where A, B or A + B is whole and the rest is a polynomial in u. (a + a sin(y)) sqrt(e cos(y)), which is elliptic, is declined before anything is asked. Numeric powers are read one level below the question too, where the polynomial term hands 1/(A B) on as (A B)^(-1), and a whole power of a product is read as the powers of its factors. Three of the file master answered in six to eleven seconds under a condition now come through in a twentieth of a second without one. 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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Rubi's
(e cos(c + d x))^p (a + a sin(c + d x))^mwith half-oddpandmran past the budget: the substitution search spent it on the two radicals. Underu = sin(y)they are(1 + u)^A (1 - u)^BwithAandBquarter-odd, which by Chebyshev's theorem is elementary only whereA + B = m + p - 1is whole.With every power a number, the rule asks only where
A,BorA + Bis whole and the rest is a polynomial inu.(a + a sin(y)) sqrt(e cos(y)), which is elliptic, is declined before anything is asked. Numeric powers are read one level below the question too, where the polynomial term passes1/(A B)on as(A B)^(-1), and a whole power of a product is read as the powers of its factors.Measured on
8a1ef454, branch against master:Re-run alone on both builds, 24 of Rubi's 4.1.1.2 go from timed out to answered. None goes the other way. The drop in families 3, 6 and 7 is problems near the budget: re-run alone, they time out on both builds or answer on both. No run had a wrong answer. Three problems in the file that master answered in 6–11 s, under a condition, now take about 0.05 s and need no condition.
FractionalPowersOfACosineBesideOnePlusASineIntegralTestdifferentiates the answers back, at both signs of the constants.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