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Integrate half an odd power of an inverse hyperbolic cosine of c + d x^2 in error functions - #1908
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Rafael-SOWNet merged 3 commits intoOct 16, 2026
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…x^2 in error functions Through u as for the negative powers: x = kappa g(u/2), and with P = a + b u the integrals of e^(+-u/2) P^(-1/2) are (2 sqrt(pi)/sqrt(2 b)) e^(-+a/(2 b)) times erfi or erf of sqrt(P)/sqrt(2 b). I(1/2) is by parts from them, and the recurrence runs up and down from I(-1/2) and I(1/2). Rubi's 7.2.5 (a + b acosh(1 + d x^2))^(5/2) and the like, from -7/2 to 5/2, were declined. The circular functions would want the Fresnel integrals, which have no node. Part of #718 Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…r-of-an-inverse-hyperbolic-cosine-of-a-quadratic
…eaking changes Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Part of #718. Follows #1904 and #1906.
#1904 and #1906 integrate whole powers of
a + b u, positive and negative, foruan inverse sine, cosine or hyperbolic cosine ofc + d x^2withc^2 = 1. This adds half an odd power for the inverse hyperbolic cosine.As in #1906,
x = kappa g(u/2)withga hyperbolic sine or cosine andkappa = x/g(u/2)constant on each side of 0, soI(m) = (kappa/2) integral g'(u/2) P^m duwithP = a + b u.g'is a sum ofe^(u/2)ande^(-u/2), and withz = sqrt(P)/sqrt(2 b):integral e^(+-u/2) P^(-1/2) du = (2 sqrt(pi)/sqrt(2 b)) e^(-+a/(2 b)) erfi(z)and... erf(z)respectively.zis written so thatz sqrt(2 b)is the root ofPitself, which keeps the branches consistent.I(1/2)is by parts from those, and the recurrence of #1904 runs up fromI(-1/2)andI(1/2), and down. The inverse sine and cosine would want the Fresnel integrals, which have no node, so they are left.Checks
APowerOfAnInverseFunctionOfAQuadraticIntegralTest: 5 more cases.Corpus, measured against master 90c9d3e (this branch then merged 0cc1e5d):
No losses. One row read as slow, 3.3:83 (145 → 620 ms). Re-timed alone, the two arms are 557–567 ms and 545–568 ms. Gate passed; suite 15,469 passed, 0 failed.
🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura