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An even power of the secant, cosine, sine or cosecant is written as the rational function of the tangent it is before the tangent substitution tests its rewriting - #1322
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…he rational function of the tangent it is before the tangent substitution tests its rewriting Timofeev's `(sec^2 - 3 tan sqrt(4 sec^2 + 5 tan^2))/(sin^2 (4 sec^2 + 5 tan^2)^(3/2))` is `((1 + u^2) - 3u sqrt(4 + 9u^2))/(u^2 (4 + 9u^2)^(3/2))` under `u = tan(x)`, two terms the algebraic partial fractions and the rational integrator answer between them, and was declined: the tangent substitution replaces `tan(x)` and asks whether an `x` survives, and the secant and the sine survived. Its answer was pinned as declined-or-right, since the general substitution had once answered it wrongly on half the line by taking `sqrt(.../cos^2)` for `.../cos`. `sec^2 = 1 + tan^2`, `cos^2 = 1/(1 + tan^2)`, `sin^2 = tan^2/(1 + tan^2)` and `csc^2 = (1 + tan^2)/tan^2`, exactly and wherever both sides are defined, and so for every even power; the tangent substitution writes them so before its test, under a root or not, since the identities hold there too. An odd power is not a function of the tangent -- its sign is not -- and is left to survive and decline as before. Rubi sample: +1 (Timofeev 596), 0 wrong, 0 timeouts. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Timofeev's
(sec^2 - 3 tan sqrt(4 sec^2 + 5 tan^2))/(sin^2 (4 sec^2 + 5 tan^2)^(3/2))is
((1 + u^2) - 3u sqrt(4 + 9u^2))/(u^2 (4 + 9u^2)^(3/2))underu = tan(x),two terms the algebraic partial fractions and the rational integrator
answer between them, and was declined: the tangent substitution replaces
tan(x)and asks whether anxsurvives, and the secant and the sinesurvived. Its answer was pinned as declined-or-right, since the general
substitution had once answered it wrongly on half the line by taking
sqrt(.../cos^2)for.../cos.sec^2 = 1 + tan^2,cos^2 = 1/(1 + tan^2),sin^2 = tan^2/(1 + tan^2)andcsc^2 = (1 + tan^2)/tan^2, exactly and wherever both sides are defined,and so for every even power; the tangent substitution writes them so
before its test, under a root or not, since the identities hold there
too. An odd power is not a function of the tangent -- its sign is not --
and is left to survive and decline as before.
Rubi sample (463 problems, 5 s budget), against the #1320 master this branch is cut from:
One more, nothing lost: Timofeev 596, in 0.7 s. No other problem moved by more than noise. Independent of #1321 (open).
The five test suites are green.
Part of #718.
🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura