A root of a square inside a sum is integrated on each side of the square's zero

[Rafael-SOWNet]

Part of #718.

1/(1 + (x^2)^(3/2)) is 1/(1 + x^3) for a positive x and 1/(1 - x^3) for a negative one. The rule for a root of a perfect square writes (x^2)^(3/2) as sgn(x) x^3 and took the sign out in front of the integral, which holds where the root is a factor of the integrand and not inside a sum: the answer was sgn(x) times the first one's, right for a positive x and wrong for every negative one. The rule that takes a square factor out of a root did the same with sqrt(x^6). 2.5.0 declined all of these. Checked with points on both sides of zero, x = ±0.4, ±1.3, ±2.7:

integrand	master 959c2d2e	this
1/(1 + (x^2)^(3/2))	right for x > 0, wrong for every x < 0	right on both sides
1/(1 + sqrt(x^2))	right for x > 0, wrong for every x < 0	right on both sides
x/(x + sqrt(x^6))	right for x > 0, wrong for every x < 0	right on both sides
1/(2 + sqrt(x^2 + 2x + 1))	right for x > -1, wrong below	right on both sides
sqrt(x^2)/(1 + x)	right on both sides	the same answer

What changes. A sign goes in front of the integral only where every occurrence of the root is a factor of the integrand, as before. Where one stands inside a sum, the root is written with a sign of its own, the integrand is integrated with it 1 and with it -1, and the answer is the piecewise of the two on the sign of the root's linear. More than one such linear, and the rule declines. A root that is a factor keeps its answer: sqrt(x^2)/(1 + x) is sgn(x) (x - ln(1 + x)) as it was.

Tests: RootOfAPerfectSquareIntegralTest.ARootOfASquareInsideASumIsIntegratedOnEachSide, seven rows, each differentiated back at four negative points and three positive ones; all seven fail on master.

Measured first with the corpus's check of the negative side switched on (IB_BOTHSIDES=1), on the 616 problems with a root of a written square in them, at the corpus's 5-second budget, against master 51cb049a, the branch then on it:

	master	this
answered	561	567
wrong	5	0

The five are 1/(1 + (x^2)^(3/2)) from 1.1.3.2 and four of 1.3.2's with sqrt(x^6) in a sum, x/(x + sqrt(x^6)) among them, each right for a positive x only. The sixth answered is one master ran out of time on.

Measured then on the Rubi corpus against master 8f3757cd, the branch's base:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1297
families 2 to 8, sampled (2410)	2215	2218

No answer is wrong in the sample on either build. The six problems the two builds disagreed on, run again one build at a time, come out the same on both: four answered and two declined.

The suite passes on the commit measured, 4667b70e, 14,598 tests of 14,611 with 13 skipped, run under a 4 GB heap limit, and so does the allocation gate. Master 959c2d2e is merged in since, without conflicts; the calculus tests pass on the merge, 4,010 of 4,011, the other the 30-second clock on declining sin(x)/(x^3 + 1)^2, which ran past it under the machine's load and passes on its own in 4 s, and every row above is as the table says on it.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura