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A root of a sum of the cosine and the sine beside a half power of one of them is read in the tangent - #1858
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…e the other way up Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…d-as-a-power-of-the-cosine
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Part of #718.
csc(x) sqrt(cos(x) + sin(x))/cos(x)^(3/2)was declined, Rubi's 4.7.7 #1123, with its respellings.SolveByTangentSubstitutionnow readssqrt(a cos + b sin)beside a half-odd power of the cosine assqrt(a + b tan)times a whole power of the cosine, and the sine the same way througha/tan + b, so what is left is a function of the tangent.The respelling is exact wherever the integrand is real: the root of a product of two non-negatives is the product of the roots, and when the sum and the cosine are both negative the two factors of
icancel. Off the real domain the answer takes the branches of the respelling.csc(x)*sqrt(cos(x) + sin(x))/cos(x)^(3/2)sqrt(cos(x) + sin(x))/cos(x)^(5/2)(1 + tan(x)) ^ (3/2) / (3/2)sqrt(2*cos(x) + 3*sin(x))/(sin(x)*cos(x)^(3/2))sec(x)*sqrt(cos(x) - sin(x))/sin(x)^(3/2)1/(cos(x)^(3/2)*sqrt(3*cos(x) + sin(x)))2 * sqrt(tan(x) + 3)Rubi corpus, branch against master's tree: one problem gained (4.7.7 #1123), none lost, 0 wrong in every family; a pocket of all of 4.7.7 gained #1122 and #1123. Kernel gate and the full suite (15261 passed) pass.
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