Integrate a trigonometric function of a + b/x beside a rational function in 1/x

[Rafael-SOWNet]

Under x = 1/u, a trigonometric function of a + b/x beside a rational function of x becomes a trigonometric function of a linear beside a rational function of u. For example, sin(c + d/x)/(e + f x) is -sin(c + d u)/(u (e u + f)) du, which the partial fractions and the sine and cosine integrals answer. x = 1/u is one-to-one on either side of 0. Every argument has to be linear in 1/x, so the question in u is not read again. The substitution search couldn't do this, because it reads a candidate's derivative but never solves the candidate for x. Below a root of a linear, sin(a + b/sqrt(c + d x)), the linear radical substitution reaches this shape first.

Corpus, measured against master 7df6694:

Set	before	this
pocket (1.3.2, 4.1.12, 1.2.2.3, 4.1.0, 4.1.7, 4.1.1.2)	2,060 / 2,160	2,072 / 2,160
families 0–8, sampled	—	equal, except family 4 +2

• 12 moves, all Unsolved → Solved: 4.1.12:267, 268, 293, 294, 321, 330, 390–392 and 395–397.
• 0 wrong, and no move the other way. The gate and suite pass.
• Exact checks: 13 rows, 2 pin sets, 4 points on both sides of 0. The residual is 0 at 50 and 2,000 digits.

The d/x rows are 0.2–2.5K chars. The rows below a root are 4.6–33K chars: the sine over linears that comes out of the substitution carries expanded seventh-degree coefficients, which is master's own defect, #1877.

Part of #718

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura