Write a polynomial in a sine over a power of a binomial in it in powers of the binomial

[Rafael-SOWNet]

Part of #718.

P(f)/(a + b f)^n, where f is a sine or cosine and P a polynomial, is written in powers of u = a + b f, with c_j = sum_i p_i C(i, j) (-a)^(i - j)/b^i. Each u^(j - n) is then integrated on its own, as a question the integrator already answers. The rule is asked after SolveAPowerOverABinomialInAPowerOfASineOverItsRoots. On master, (A + B cos + C cos^2)/(a + b cos)^4 was answered in 30 million characters after 18 s; it is now 9K characters in about 1 s.

Rubi chain (master arm at 1e79b0e, 5 s budget):

• Pocket (4.1.3.1, 4.1.4.x, 4.2.3.1, 4.2.4.x; 1,490 rows): master 1,475, this branch 1,488.
• Families 0–8 sampled: identical on both arms.
• Moved rows run alone: master solves 0 of 13, the branch 13 of 13.
• The gate and the suite (15,495 passed) both pass. Exact checks at 50 and 300 digits: 72 of 72.

On 2.5.0 all thirteen were integral(...), some after 27 s. The branch answers them in 1.2K to 10.9K characters, in 0.5 to 3.8 s each.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura