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Integrate a power of x times the logarithm of a polynomial plus a root, by parts - #1955
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Rafael-SOWNet merged 1 commit intoOct 20, 2026
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…t, by parts With D = A^2 - B^2 Q, the derivative of ln(A + B sqrt(Q)) is D'/(2 D) + (Q (A B' - A' B) + A B Q'/2)/(D sqrt(Q)), an identity of the algebra with no branch in it, so by parts what is left is a rational function and one over the root, which the rules for those answer. The search differentiated the logarithm as written, a quotient with the root in both parts, and ran past the budget. Only for a D that is not constant: the inverse hyperbolic functions, whose D is, are answered already. Rubi's 3.5: rows 136, 143, 146 and 147 go from unevaluated or timed out to solved, 0 wrong. Families 0 and 1 and samples of 2-8 otherwise identical; gate and suite pass. 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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Part of #233.
x^3 ln(-1 + 4 x + 4 sqrt((x - 1) x))was declined, and three more of Rubi's 3.5 with half-odd powers ofxbeside the same logarithm ran out of time. LetD = A^2 - B^2 Q. The derivative ofln(A + B sqrt(Q))isD'/(2 D) + (Q (A B' - A' B) + A B Q'/2)/(D sqrt(Q)), an identity of the algebra with no branch in it. By parts, what is left is a rational function plus one over the root, and the rules for those answer it. HereD = 8 x + 1.The new rule is
SolveByPartsAgainstALogarithmOfARadicalSum. It applies only for aDthat is not constant. The inverse hyperbolic functions have a constantD, and they are answered already:x ln(x + sqrt(x^2 + 1))is left to its existing route.Measured against master
8a1ef454:The rebase onto #1950–#1954 conflicted only on the dispatch line, at the shared anchor. I kept both. The combined tests of #1952, #1953, #1954 and this branch pass, 17/17. All four rows were declined by 2.5.0, and
BREAKING-CHANGES.mdhas the entry.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura