Integrate a symbolic power of a trigonometric function where it is the derivative of a product

[Rafael-SOWNet]

Part of #718.

f^m (alpha + beta f^2), for a symbolic m and f a sine, cosine, secant or cosecant of a linear u, is the derivative of a product where its bracket is a multiple of that derivative's. With u' = 1:

f	product	its derivative
sin	cos sin^(m + 1)	sin^m ((m + 1) - (m + 2) sin^2)
cos	sin cos^(m + 1)	cos^m ((m + 2) cos^2 - (m + 1))
sec	tan sec^m	sec^m ((m + 1) sec^2 - m)
csc	cot csc^m	-csc^m ((m + 1) csc^2 - m)

SolveASymbolicPowerOfATrigonometricFunctionAsTheDerivativeOfAProduct reads the integrand as f^m times a polynomial in f with only a constant and a square term. It matches alpha + beta f^2 against p0 + p2 f^2 where beta p0 = alpha p2, and answers (alpha/p0) G/u'. The match is exact, with nothing sampled. SolveAsTheDerivativeOfAProductOfPowers raises the powers an integrand has, and the sine above the bar that tan sec^m brings is not one of them.

The power is taken off by writing 1 for it rather than dividing by it, because s^m (...)/s^m is not cancelled by the simplification.

Checks

• Exact checks at 50 and 300 digits on seven shapes, with m and n of both signs: 126/126.
• ASymbolicPowerOfATrigonometricFunctionAsADerivativeIntegralTest: 6 cases, among them one whose bracket is not the derivative and is not claimed.

Corpus, measured against master 90c9d3e (this branch then merged bc175c2):

	master	this branch
pocket: every row of the 4.x.4.x files (1,821 statable)	1,798	1,804
families 0–8 samples	4.1.4.1:8, 4.5.4.1:23, 38	+3 (the same rows)
rows moved		+6: 4.1.4.1:8; 4.2.4.1:50, 51; 4.5.4.1:23, 24, 38

All six solve on this branch re-run alone and none on master. No losses. Three 4.x.4.2 rows read as slow. Re-timed alone twice per arm, they are 790–854 vs 821–881 ms, 220–245 vs 217–233 ms and 268–270 vs 266–285 ms. Gate passed; suite 15,470 passed, 0 failed.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura