An even power of the secant or the cosecant under a root is written in the tangent - #1717
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…n the tangent The tangent substitution writes sec^2 as 1 + tan^2 and csc^2 as (1 + tan^2)/tan^2, but went on only where the tangent, or a sine or a cosine under a root, was already in the integrand, so sqrt(a + b csc(x)^2) was declined where sqrt(a + b cot(x)^2) was answered. An even power of the secant or the cosecant under a root goes on now too. Rubi's 4.5.7 and 4.6.7. Part of #718 Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…e-secant-under-a-root
…e-secant-under-a-root
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Part of #718.
sqrt(a + b csc(x)^2)was declined wheresqrt(a + b cot(x)^2)was answered, although underu = tan(x)the two are one integrand:csc^2is(1 + tan^2)/tan^2.8f3757cdsqrt(a + b csc(x)^2)(a + b csc(c + d x)^2)^(3/2)sqrt(a + b sec(x)^2)1/sqrt(a + b sec(x)^2)sqrt(1 + csc(x)^2)1/sqrt(-1 + csc(x)^2)What changes. The tangent substitution writes an even power of the secant as
1 + tan^2and of the cosecant as(1 + tan^2)/tan^2, but it went on only where the tangent, or a sine or a cosine under a root, was already in the integrand. It goes on now for an even power of the secant or the cosecant under a root too. The cosecant's answers carrysgn(tan(x)), from the root of1/tan(x)^2.Tests:
TangentSubstitutionIntegralTest.AnEvenPowerOfTheSecantOrTheCosecantUnderARoot, eight rows, each differentiated back.Measured first on the 124 problems of 4.5.0, 4.5.7, 4.6.0, 4.6.7 and 4.7.7 with an even power of the secant or the cosecant under a root, at the corpus's 5-second budget, against master
dbe5b382, the branch's base:Measured then on the Rubi corpus against master
dbe5b382:No answer is wrong on either build. The 76 problems the two builds disagreed on, run again one build at a time, are 74 answered here and not on master -- 52 in 4.5.7, 19 in 4.6.7, Timofeev's 591 and 592,
(4 - 5 sec(x)^2)^(3/2)and its reciprocal, and one in 4.7.7 -- and two that are the machine: one both builds answer alone, one both run out of time on.The corpus was measured on
6d5dce68. Master8f3757cdis merged in since, without conflicts, and the suite passes on that merge,107efc7c, 14,599 tests, and so does the allocation gate.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura