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Integrate a rational function of the tangent beside roots of linears in it - #1970
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Rafael-SOWNet merged 1 commit intoOct 20, 2026
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…in it Under t = tan(z), dz = dt/(f (1 + t^2)), and a rational function of the tangent beside roots of linears in it is a rational function of t beside roots of linears in t. It is now integrated term by term over the partial fractions of its rational part, with the 1 + t^2 below the bar written as the linears of its roots, (1 + i t)(1 - i t). Each term is the roots over one power of a linear, which the rules for a root of a linear over a linear answer. Rubi's 4.3.4.2 (c + d tan(z))^(5/2) (A + B tan(z) + C tan(z)^2)/sqrt(a + b tan(z)) and six more of its rows, and three of 4.3.2.1, were past the budget. 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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Under
t = tan(z),dz = dt/(f (1 + t^2)). A rational function of the tangent beside roots of linears in it is therefore a rational function oftbeside roots of linears int.It is now integrated term by term over the partial fractions of its rational part. The
1 + t^2below the bar is written as the linears of its roots,(1 + i t)(1 - i t), so each term is the roots over one power of a linear, which the rules for a root of a linear over a linear answer.Measured on
8a1ef454, branch against master:Re-run alone on both builds, 10 problems go from timed out to answered, in 0.2–3 s: Rubi's 4.3.2.1:1329/1334/1335 and 4.3.4.2:181/182/188/194/198/199/204. None goes the other way. No run had a wrong answer.
ARationalFunctionOfTheTangentBesideRootsOfLinearsInItIntegralTestdifferentiates 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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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura