Integrate a rational function times a logarithm of powers of linears by parts

[Rafael-SOWNet]

Part of #718.

integral R ln(G), with R a rational function and G a product of powers of linears nested in powers and constants, is now integrated by parts: S ln(G) - integral S G'/G, with S = integral R. G'/G is read off G as it is written: a product's is the sum of its factors', a power's is the exponent times its base's, and a linear's is b/(a + b x). That is exact by the chain rule on any branch, and rational. Rubi's 3.2.3 ln(e (f (a + b x)^p (c + d x)^q)^r) was declined.

SolveARationalFunctionTimesALogarithmOfPowersOfLinears:

• needs S to come back rational, so the remainder stays rational;
• asks the remainder as the same question rather than one of its own, because lifted it may not lift again, and the sum it splits into was declined;
• is not checked at sampled points: each step is exact by construction, and the check reported a difference for an answer that agrees with the integrand to fifty digits at the very points it pins (ln(k (f (a + b x)^p (c + d x)^q)^r) with p = 5.37, r = 2.83).

Fenced after the first chain. Asked first with a polynomial beside the logarithm, it pre-empted by parts: (f + g x)^4 (A + B ln(e ((a + b x)/(c + d x))^n)) ran out of time on its remainder, and (g + h x) ln(...) slowed from 1.2 s to 3.6 s. A polynomial of degree one or more is now asked only once everything else has declined, from a second, late call. That keeps x ln(...), which nothing else answers.

Checks

• Exact checks at 50 and 300 digits on five shapes, three pin sets: 120/120.
• ARationalFunctionTimesALogarithmOfPowersOfLinearsIntegralTest: 4 cases.

Corpus, measured against master 90c9d3e (this branch then merged 87be872):

	master	this branch, fenced, re-run back to back
the moved rows and 3.2.3:45 and :30, 16 in all	8	13
rows gained		+5: 3.2.3:30, 46, 49; 3.3:610, 612

The chain itself ran on all of family 3 before the fence (3,085 rows; master 1,340 of 1,382 statable, branch 1,343). The fence came from that run, and the rows it moved were re-run on both arms afterwards. Gate passed before the fence; the suite on the fenced branch passed: 15,468, 0 failed.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura