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Integrate the incomplete gamma function of a whole order by parts against a reciprocal power - #2082
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Rafael-SOWNet merged 1 commit intoOct 23, 2026
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…inst a reciprocal power Gamma(n, u) for u = a + b x is now a factor that integration by parts differentiates first, as the logarithms and the other special functions are: its derivative -u^(n - 1) e^(-u) u' is elementary. Against 1/(c + d x)^k with k >= 2 and a whole n <= 0, what is left is (a + b x)^(n - 1) e^(-a - b x) over a power of c + d x, a rational function times an exponential, which partial fractions and the exponential integral answer. Rubi's 8.6, read with the harness copy that maps GAMMA: 144 -> 156 of 177, the twelve Gamma(n <= 0, a + b x)/(c + d x)^k rows, 0 wrong, no other row moved, each in 0.15 to 0.32 s. No integrand in the rest of the corpus holds the function. 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
gamma(n, a + b x)of a whole ordern <= 0over a power of another linear, by parts. Part of #2075.The function is now a factor that integration by parts differentiates first, as the logarithms and the other special functions are: its derivative
-u^(n - 1) e^(-u) u'is elementary. Against1/(c + d x)^kwithk >= 2, what is left is(a + b x)^(n - 1) e^(-a - b x)over a power ofc + d x. That is a rational function times an exponential, which partial fractions and the exponential integral answer. One arm inIsDifferentiatedBeforeAPolynomial.aff8368d's integration rules, #2078)Read with the harness copy that maps Maxima's
GAMMA. The 12 rows are exactly theGamma(n <= 0, a + b x)/(c + d x)^kones, each answered in 0.15 to 0.32 s and 16 to 71 MB. No other row moved, and no integrand in the rest of the corpus holds the function.The suite passes (16,894) and so do the MCP tests (329), measured before rebasing onto #2080, which touches nothing this reads. The new tests differentiate four of the rows back.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura