An odd power of a sine or cosine of a logarithm is integrated; a resonance on a sign is declined

[Rafael-SOWNet]

Part of #718. Fixes #1852.

An odd power of a sine or a cosine of a logarithm is integrated. The substitution t = ln(c x^n) turns x^m sin(a + b ln(c x^n))^3 into an exponential times powers of sines and cosines. A logarithm's substitution now asks that integrand as a question of its own, from the top, so the top-only rule for it (#1265) answers. A thread-static flag stops the question lifting twice, and a decline under the flag is scoped.

A resonance whose vanishing exponent rests on a sign is declined (#1852). In exp(x/n) sin(x sqrt(-1/n^2)), 1/n + i sqrt(-1/n^2) is zero when n > 0, and 1/n - i sqrt(-1/n^2) is zero when n < 0. The resonant branch divided by whichever factor it could not show to be zero, so the answer had no value anywhere. That is the answer on 2.5.0 and on master. The branch now declines unless one of the two factors is shown to vanish. A numeric resonance such as e^(2x) sin(x sqrt(-4)) is still answered.

	2.5.0	master 3f651b4	branch
sin(ln x)^3	declined	declined	130 chars
sin(a + b ln x)^3	declined	declined	909 chars
x^2 sin(a + b ln(k x^n))^3	declined	declined	1,605 chars
x^m cos(a + b ln(k x^n))^3	declined	declined	1,907 chars
sin(a + b ln x)^4 / x^2	declined	> 300 s	2,083 chars
exp(x/n) sin(x sqrt(-1/n^2))	an answer with no value	an answer with no value	declined

Measured

• Rubi 4.5–4.7, full files: 4569 solved on master against 4584 on the branch, with 0 wrong. The 15 gained rows are all real-b cube rows in 4.7.5.
• The 25 imaginary-b rows of 4.7.5: they decline, as on master. Before the fix, they were the 23 wrong answers through A resonant exponential times a sine with a symbolic sign divides by zero #1852.
• Families 0–8 against master 3f651b4: 0 rows moved and 0 wrong.
• Gate: passed.
• Suite: 15,243 passed, 0 failed.

The test differentiates each answer back at four points and bounds its size. Three resonant rows must stay unevaluated.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura