A radical function of whole powers of x written through its logarithm says where its answer holds

[Rafael-SOWNet]

Part of #718.

1/csch(2 ln(c x))^(1/2) is sqrt(sinh(2 ln(c x))), which is sqrt(((c x)^2 - (c x)^(-2))/2) and real on both sides of zero. Master integrates it under t = ln(c x), and the answer is right for a positive c x and wrong for every negative one; 2.5.0 declined it. Checked on both sides of zero at c = 1.3, x = ±1.3, ±1.9, ±2.7:

integrand	master b2157e08	this
1/csch(2 ln(c x))^(1/2)	right at the three positive points, wrong at the three negative ones	the same answer provided c x > 0, right at the three positive points
csch(2 ln(x))^(-3/2)	the same	the same, provided x > 0
1/sech(2 ln(x))^(1/2)	the same	the same, provided x > 0
sqrt(sinh(2 ln(c x)))	right at all six	unchanged

Why. For a negative c x, t = ln(c x) is ln(-c x) + i pi. The integral in t is found by rules that take t to be real: the exponential substitution u = e^t takes u out from under a root as positive. A function of whole powers of e^t is real at t + i pi as well, so for those integrands it shows: the answer is the integrand's antiderivative for a positive c x and not for a negative one.

What changes. Where the integrand in t holds t only in exponentials e^(k t + a) with a whole k, some of them under a power that is not whole, the logarithm's two substitution rules give their answer provided c x^n > 0 -- or x > 0 -- and nothing is said for a negative c x^n. For an even n and a positive numeric c the condition always holds and is left off. A rational function of e^t uses no positivity and keeps its answer as it was. The same integrand written in powers of x, sqrt(sinh(2 ln(c x))) above, is answered on both sides by the fold of an exponential of a logarithm before either rule, as before.

Tests: LogarithmSubstitutionTest.ARadicalFunctionOfAWholePowerOfX, six rows compared at three negative and three positive points: four answered provided the logarithm's argument is positive, and holding at every positive point, which fail on master at the negative ones; and two written in powers of x, which hold at all six.

Measured first with the corpus's check of the negative side switched on (IB_BOTHSIDES=1) on the 177 problems with an exponential or a hyperbolic function and a logarithm in them, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
answered	149	155
wrong	6	0

The six wrong on master are 6.5.3's 1/sech(2 ln(c x))^(1/2) and ^(3/2), and 6.6.3's 1/csch(2 ln(c x))^(1/2), ^(3/2), csch(2 ln(c x))^(1/2)/x^4 and csch(2 ln(c x))^(3/2)/x^2, each right for a positive x only; here all six are answered provided c x > 0. Two of them are checked only after Simplify() on both builds, since the derivative of the raw answer holds a sgn the harness does not evaluate.

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1297
families 2 to 8, sampled (2410)	2215	2219

No answer is wrong in the sample on either build. The difference is five problems far past the corpus's five seconds on both builds, whose verdict there is decided by when the budget is noticed: run again one build at a time, three of them went the other way. Measured alternately with no budget, four are answered on both builds in the same time, 16 to 35 s -- log(e (f (a + b x)^p (c + d x)^q)^r)/(a + b x)^4, sqrt(g sin(e + f x))/((c + d sin(e + f x)) sqrt(a + a sin(e + f x))), x/csch(x)^(5/2) + 3/5 x/sqrt(csch(x)) and 1/(x (a x + b x^3 + c x^5)^2) -- and 1/((a g + b g x)^3 (A + B ln(e (a + b x)/(c + d x)))) runs past 150 s on both.

The suite passes on the commit measured, 537ca738, 14,597 tests with 13 skipped, run under a 4 GB heap limit, and so does the allocation gate. Master b2157e08 is merged in since, without conflicts; the calculus tests pass on the merge, 4,017 of them, and every row above is as the table says on it.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura