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Undo an inverse sine or cosine to a power that is not whole - #2100
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Rafael-SOWNet merged 1 commit intoOct 23, 2026
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The substitution that undoes an inverse trigonometric function was taken for a radical it removes, an exponential of the inverse, or a whole power of it above the first, and left a first power to parts. A power that is not whole, arcsin(a x)^n, went to neither, and parts cannot lower it. Under the sine, x^2 arcsin(a x)^n is u^n sin(u)^2 cos(u)/a^3, Euler's formula's exponentials of i u beside a power of u, each a gamma function; so it is now taken for such a power of an inverse sine or cosine. Not of the others: 1/((c + a^2 c x^2) sqrt(arctan(a x))) is a derivative of the inverse's root, answered in 0.7 s elsewhere, and under the tangent it took 5 s and 1.6 GB. 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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Takes the substitution that undoes an inverse sine or cosine for a power of it that is not whole. Part of #2075.
The rule. The substitution that undoes an inverse trigonometric function was taken in three cases: for a radical it removes, for an exponential of the inverse, or for a whole power of it above the first. A first power was left to parts. A power that is not whole,
arcsin(a x)^n, fell under none of them, and parts cannot lower it.Under the sine,
x^2 arcsin(a x)^nisu^n sin(u)^2 cos(u)/a^3. Euler's formula writes that as exponentials ofi ubeside a power ofu, and each term is a gamma function (#2085). The substitution is now taken for such a power of an inverse sine or cosine.It is not taken for the other inverse functions. The first version took them too, and
1/((c + a^2 c x^2) sqrt(arctan(a x)))then took 5 s and 1.6 GB under the tangent. That integrand is the derivative of the inverse's root, and another route answers it in 0.7 s and 59 MB.Measured on Rubi's suite, with the harness that reads
GAMMA, against68211405, the whole corpus each side. This branch was measured on2be40bad, which adds #2088 to that base, and is rebased onto master now.68211405The gains are 11 rows, the last 5.1.2, 5.2.2 and 5.1.5 had:
5.1.2:157-160,5.2.2:157-160and5.1.5:260-262. The other 5 are #2088's. Family 5 was run again after the restriction, all 1,797 rows: every row solved, the same 11 gains, 0 wrong. The arctan rows' allocation is back to master's, 49 MB each. One row's allocation grew in that run,5.6.2:38, which is the same alone on both builds (0.48 GB).Four integrals differentiate back in the tests. The suite passes (16,956), and so do the MCP tests (329).
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura