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Divide a polynomial over a polynomial beside a power of a quadratic without the quadratic's condition - #1985
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Rafael-SOWNet merged 1 commit intoOct 21, 2026
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…ithout the quadratic's condition tan^2/(a + b tan + c tan^2)^(3/2) is x^2/((1 + x^2) Q^(3/2)) under the tangent. Partial fractions divided the numerator by the whole denominator, Q^(3/2) with it, and left 1/Q^(3/2), which is answered, and a remainder -1/((1 + x^2) Q^(3/2)) provided not Q = 0, which nothing read; without the condition it is answered. The condition holds wherever the integrand is defined, so a new rule at the end of the dispatch asks the remainder again without conditions of the form not F = 0 on a factor of the denominator, and answers only where the quotient and the remainder are both answered. Rubi's 4.3.9:39 is answered, the only unsolved row of 4.3.9 and 4.4.9, in 3,042 characters, the size of master's tan and tan^3 neighbours. 233 solved neighbours, all of 4.3.9 and 4.4.9 among them, answer as before, character for character. 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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tan^2/(a + b tan + c tan^2)^(3/2)isx^2/((1 + x^2) Q^(3/2))under the tangent. Partial fractions divided the numerator by the whole denominator,Q^(3/2)with it. That left1/Q^(3/2), which is answered, and a remainder-1/((1 + x^2) Q^(3/2))carryingprovided not Q = 0, which nothing read. Without the condition, the remainder is answered.The condition holds wherever the integrand is defined. So a new rule,
SolveByDividingWithoutTheConditionsOfTheDenominator, at the very end of the dispatch and at the question asked or one below, asks the remainder again without it. It requires a polynomial numerator, an improper fraction and a quotient that is not zero. It drops only conditions of the formnot F = 0whereFis a factor of the denominator, or the base of a positive power of one, and declines on any other condition. It answers only where the quotient and the remainder are both answered. Retrying inside the division instead lengthened four solved rows of 4.3.9 and 4.4.9 by 4 to 198 characters, so the rule sits at the end.Measured on
8270086f, branch against master:Re-run alone, 4.3.9:39 goes from unevaluated to answered, in 3,042 characters, the size of master's answers for its
tanandtan^3neighbours. It is the only unsolved row of 4.3.9 and 4.4.9. The other two moves, 3.2.1:292 and 7.4.1:369, are answered on both builds.Of 233 solved neighbours, every one answers as before, character for character. They include all of 4.3.9 and 4.4.9 and 160 quotients with a root from families 1 and 4.3–4.6. Quadrature agrees on [0.1, 0.9], and the answer was checked at four pin sets on both sides of the tangent's poles. None of the problems answered on both builds allocates more on the branch. No run had a wrong answer.
DividingWithoutTheConditionsOfTheDenominatorIntegralTestdifferentiates the answers back.BREAKING-CHANGES.mdhas the entry, with the 2.5.0 column measured on v2.5.0.Part of #233.
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