Integrate a power of an inverse function of c + d x^2 by its recurrence

[Rafael-SOWNet]

Part of #718.

For u the inverse sine or cosine of w = c + d x^2 with c^2 = 1, or its inverse hyperbolic cosine with c^2 = 1, (a + b u)^n for a whole n from 1 to 12 has an elementary antiderivative by a recurrence that closes at the first two powers:

integral (a + b u)^n = x (a + b u)^n + 2 e n b H (a + b u)^(n - 1) - 4 e b^2 n (n - 1) integral (a + b u)^(n - 2)

Here H = 1/(d x u'(w)), and e is 1 for the circular functions and -1 for the hyperbolic one. Each u has u'' = e w u'^3, and with c^2 = 1 the radicand of u' is d x^2 (2 c + d x^2) up to sign. So H u' = 2, and the derivative of the first two terms is (a + b u)^n plus a multiple of (a + b u)^(n - 2). The parser writes acosh(w) as ln(w + sqrt(w^2 - 1)), and the rule reads it from that form (and asinh with c^2 = -1 the same way).

SolveAPowerOfAnInverseFunctionOfAQuadraticByTheRecurrence checks H u' = 2 symbolically, with the derivative the library takes, rather than at sampled points. The sampled check declined the right answer to asin(1 + d x^2)^2. With c = 1 and the positive d it pins, w is past 1, on the cut of the inverse sine.

Checks

• Exact checks at 50–2000 digits on nine shapes, each with d of both signs, on each real domain and off it: 432/432.
• APowerOfAnInverseFunctionOfAQuadraticIntegralTest: 8 cases.

Corpus, measured against master 55a8044 (this branch then merged c92dbe0):

	master	this branch
pocket: every row of 5.1.5, 5.2.5, 7.1.5, 7.2.5 (719 statable)	642	668
families 0–8 samples	5.2.5 and 7.2.5 rows	+3 (the same rows)
rows moved		+26: 5.1.5:534–536, 541–543, 548, 549; 5.2.5:136–138, 143–145; 7.1.5:415–417, 422–424; 7.2.5:343–345, 350–352

Twelve of the 26 were timeouts on master; on this branch the slowest takes 0.47 s. No losses and no slower rows. Gate passed. The chain's suite aborted with the test host exiting and no exception logged, 14,372 tests in. Re-run alone it passed: 15,421 passed, 0 failed.

acos(1 + d x^2) was answered on master in 106 characters and is 76 here; 2.5.0 declined it.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura