A power of the variable times a sine or cosine of a logarithm, and a power of a monomial

[Rafael-SOWNet]

x^2 sin(a + b ln(c x^n)) and (c x^n)^b were both left as written. Two rules, and both of their side conditions were written by a measurement rather than by foresight.

The trigonometric of a logarithm is a closed form. Two rounds of parts close on the integrand — the sine's remainder is the cosine's integral, the cosine's is the sine's — so the pair is solved rather than iterated: int x^m sin(L) dx is x^(m+1)((m+1) sin(L) - B cos(L))/((m+1)^2 + B^2) wherever L' = B/x, which a + b ln(c x^n) is with B = b n. Differentiating the answer is the whole proof. The hyperbolic twin needs no rule at all — the library writes sinh as exponentials, which fold against the logarithm (#1470) — while sin and cos are nodes that fold against nothing.

A power of a monomial is distributed, (c x^n)^b → c^b x^(n b), for a positive c and a non-whole n. Both conditions are the domain's: with a fractional or symbolic n the integrand is real only where x > 0, and there the identity is exact; with a whole n it is real at a negative x too, and there the value is a power of |x|.

What the measurement caught, in the order it caught it:

• The first version had no condition on n, and family 4 leapt +13 — all of it unsound. At depth, under u = tan(x), the rule saw (2 u^3)^(3/2) and distributed it to 2^(3/2) u^(9/2), dropping the sgn(u) that the root-of-an-even-power rules write. The suite's own ARootOfAnEvenPowerIsNotThePower failed on exactly that integrand. With the condition, the gain is +1 and the sign is back.
• The same run reported 1 wrong answer: x^2 sin(a + ln(c x^n) sqrt(-9/n^2)), where the closed form's divisor (m+1)^2 + B^2 is 9 + (n sqrt(-9/n^2))^2 — a zero that only Simplify folds. An imaginary B of exactly that size makes the two rounds circular rather than closing, so the "closed form" divided by nothing at all. It is declined now, and the divisor is tested symbolically.

Measured (final build):

• Family 4 (familycap.sh 4 6): 289 → 290/422, 0 wrong, 0 error — 4.7.5 goes 2 → 3 of its 6.
• Family 1: 158/228, unchanged. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,671 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• Seven forms verified by differentiating back at five points — numeric, symbolic and fully symbolic (x^m cos(a + b ln(c x^n))) — worst relative deviation 0.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura