Integrate a power of a sine or hyperbolic sine over a binomial in a power of it over its roots

[Rafael-SOWNet]

Part of #718. Extends #1905.

#1905 splits 1/(a + b s^n) over the roots of the binomial, for s a sine, cosine or hyperbolic one. This takes a power of s above or below the bar as well:

• k ≥ n: s^k/(a + b s^n) is s^(k - n)/b - (a/b) s^(k - n)/(a + b s^n).
• k < 0: it is s^k/a - (b/a) s^(k + n)/(a + b s^n).
• 0 < k < n: it is (1/(n a)) sum r^(k + 1)/(r - s) over all n roots.

So each term is the per-root closed form #1905 already writes, and the powers of s beside go to the integrator. It is asked early, where a power stands above or below: the rules before it spent the budget on sinh(y)^4/(a + b sinh(y)^3). The bare reciprocal is left to the reading #1905 gives it, and its answers are unchanged.

Fenced after the chain. At n = 4 with a power beside it, the rules before it answer sin(y)^3/(a - b sin(y)^4) in a tenth of a second, and asked first this took half of one. n = 4 is now theirs. The 33 rows the chain moved or slowed were re-run on both arms after the fence: the nine gains, and nothing slower.

DerivativeDiffersWhereReal decides a difference again in decimals. It counted a point where the answer's derivative had no value at the default precision as a difference. The derivative of the right answer to sinh(c + d x)^5/(a + b sinh(c + d x)^3) is NaN there at x = 3.17 with the pinned symbols, and the answer was refused. A difference, or no value, at the default precision is now decided again at forty and then eighty digits:

• still no value where the integrand has one is still a difference;
• a precision that cannot tell says nothing.

Checks

• Exact checks at 50 and 300 digits on eleven shapes, three pin sets: 198/198.
• AReciprocalOfABinomialInAPowerOfASineIntegralTest: 4 more cases.

Corpus, measured against master 90c9d3e (this branch then merged bf4be60):

	master	this branch
pocket: the algebraic pocket and every row of 4.1.7, 4.2.7, 6.1.7, 6.2.7 (8,745 statable)	8,696	8,705 before the fence
the 33 rows moved or slowed, re-run on both arms after the fence	23	32
rows gained		+9: 6.1.7:200–204, 207–210

No losses. Gate passed; the suite on the fenced branch passed: 15,468, 0 failed.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura