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Integrate a power of a logarithm beside a linear in another as two powers of the first - #2094
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…wers of the first (a + b L)^p (d + e M), with L = ln(c x^n) and M = ln(f x^r), was declined, and so was it with M = L. M is (r/n) L + delta, where delta = M - (r/n) L is constant between the cuts: its derivative is r/x - (r/n)(n/x) = 0. So d + e M is alpha (a + b L) + beta with alpha = e r/(n b), and the integrand is alpha (a + b L)^(p + 1) + beta (a + b L)^p beside what else is there, each answered by the rules for a power of one logarithm. delta is written back in the answer. Asked after the product of powers read as a derivative, which writes p x^m ln(x)^n ((m + 1) ln(x) + n + 1) as p x^(m + 1) ln(x)^(n + 1), where this would write a sum of gamma functions. Part of #2075. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Integrates a power of a logarithm of a power of x beside a linear in another such logarithm, as two powers of the first. Part of #2075.
The rule. For
(a + b L)^p (d + e M)withL = ln(c x^n)andM = ln(f x^r),Mis(r/n) L + delta. Heredelta = M - (r/n) Lis constant between the cuts, because its derivative isr/x - (r/n)(n/x) = 0. Sod + e Misalpha (a + b L) + beta, withalpha = e r/(n b)andbeta = d + e delta - alpha a. The integrand is thenalpha (a + b L)^(p + 1) + beta (a + b L)^pbeside whatever else is there. Each of those is a power of one logarithm, which the existing rules answer, anddeltais written back into the answer.M = Lis the same rule withdelta = 0.It is asked after the product of powers read as a derivative. The first version came before it, and answered
p x^m ln(x)^n ((m + 1) ln(x) + n + 1)as a sum of gamma functions, where that rule writesp x^(m + 1) ln(x)^(n + 1). A test that pins that form failed.Measured on Rubi's suite, with the harness that reads
GAMMA, against68211405, the whole corpus each side. This branch sits on2be40bad, which adds #2088 to that base.68211405The gains are 3.1.5's 7 rows,
3.1.5:219-226. The other 5 are #2088's:6.1.1:205-207and6.1.3:98/100. Family 3 was run again after the move, 1,481 rows: the same 7 gains, 0 wrong, 0 lost. Allocation per stream is 1.0003, 1.0001 and 0.9918 of the base's, and time 1.087, 1.037 and 0.950, with other builds running. Three rows' allocation grew in the run:3.3:246,6.1.7:196and8.1:296. Alone they are the same on both builds: 2.46, 1.07 and 1.39 GB.Four integrals differentiate back in the tests. The suite passes (16,956), and so do the MCP tests (329).
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura