A term with a power of x is integrated by parts with the rest of the sum in its remainder

[Rafael-SOWNet]

Part of #718.

sqrt(csc(x)) (x cos(x) - 4 sec(x) tan(x)), Rubi's 4.7.7 #1129, is (2 x sqrt(sin(x)) - 4 sin(x)^(3/2)/cos(x))', and neither of its terms is elementary alone. Where a sum's terms are not integrated one by one, at the question asked, a term x^k f is integrated by parts against x^k. The leftover, -k x^(k-1) F, is then asked together with the other terms. If that leftover is zero, the parts are the answer, as for (x^2 e^sin(x))'. When the leftover is a sum of c sin^a cos^b over one argument, the least powers are taken out. It closes as K sin^(a+1) cos^(b+1) when it is one.

Each limit comes from a measurement. On the first corpus run, the suite hung on (c + d x) erfc(a + b x)^2. (x^2 + 3)^2/(x + sqrt(x^2 + 3)) went from a 0.5 s decline to past two minutes, and (a + b atanh(c x^2))^2/x^5 doubled to 37 s:

• the rule does not fire inside its own questions;
• f must have x only inside transcendental functions;
• a term whose F keeps a polynomial in x goes into the remainder untouched, since asking that remainder again would not get any simpler.

Measured on a4993e6 (master) and on 2.5.0:

Integrand	2.5.0	master a4993e6	this branch
sqrt(csc(x))*(x*cos(x) - 4*sec(x)*tan(x))	declined	declined	72 characters
x*cos(x)/sqrt(sin(x)) - 4*sqrt(sin(x))/cos(x)^2	declined	declined	61
x*sin(x)/sqrt(cos(x)) + 4*sqrt(cos(x))/sin(x)^2	declined	declined	62
sqrt(csc(2*x + 1))*(x*cos(2*x + 1) - 2*sec(2*x + 1)*tan(2*x + 1))	declined	declined	94
x*cos(x)*e^sin(x) + e^sin(x)	declined	declined	x * e ^ sin(x)
x^2*cos(x)*e^sin(x) + 2*x*e^sin(x)	declined	declined	x ^ 2 * e ^ sin(x)
(c + d*x)*erfc(a + b*x)^2	erfc read as a symbol	declined, 13 s	1,196 characters, 7 s; differentiates back in SpecialFunctionsByPartsTest

Corpus chain against master a4993e6:

• 4.7.7 pocket plus families 0–8: one move, 1129 gained; 0 wrong answers.
• Allocation gate: passed.
• Suite: 15,271 passed.

The every-target build succeeds with 0 warnings.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura