Two roots of linears over a linear off the real line are integrated

[Rafael-SOWNet]

Part of #718.

1/(sqrt(x) sqrt(a + b x) (1 - i x)) was answered on master with a piecewise that held at no point. 2.5.0 declined it:

integrand	2.5.0	master 092b2815	this
1/(sqrt(x) sqrt(a + b x) (1 - i x))	declined	no value at any point	99 characters
sqrt(x)/(sqrt(a + b x) (1 + i x))	declined	no value at any point	442 characters
sqrt(x) sqrt(a + b x)/(1 + i x)	declined	no value at any point	597 characters

Each answer is differentiated back and compared with the integrand as complex numbers at six points.

What changes. SolveAPolynomialOverAPowerOfALinearBesideTwoRoots ends with two closed forms over the roots of two linears, one for int 1/S and one for int 1/(y S) with y a third linear, each an arctangent for one sign of a quantity and a logarithm for the other, chosen by BySign: the form for the sign of a number, and both, each where it holds, for a quantity with symbols in it. Over a linear whose coefficients are not real the quantity has no sign -- -i a - b < 0 is NaN, the complex numbers not being ordered -- and neither arm held. Both forms use nothing about their root but sqrt(q)^2 = q, so either is an antiderivative wherever the quantity is not zero, whatever its phase; the sign only chooses the form that is real on the real line. The two calls say so now, and a quantity with the imaginary unit in it takes the form for a positive one alone, rather than a piecewise. The other callers of BySign, an arcsine among their forms, need more of their root and are left as they were.

Tests: RootsOfLinearsOffTheRealLineIntegralTest, three rows compared as complex numbers; master answers all three with a piecewise that has no value at any of the four points.

Measured first on all of Rubi's 4.3.2.1 and 4.3.3.1, 1,948 problems run, at the corpus's 5-second budget, against master 287c69a7, the branch's base:

	master	this
solved	1514	1515
wrong	0	0
past the budget	308	308

Measured then on the Rubi corpus against master 287c69a7:

	master	this
family 0, independent suites (1814)	1772	1772
family 1, 40 a file (1381)	1308	1306
families 2 to 8, sampled (2410)	2301	2301

The corpus cannot see the change: an answer with no value at any point is filed by the harness as one it cannot check on the reals, and so is one compared as complex numbers, so master's piecewise and this branch's answers count alike. The seven problems the two builds disagreed on, run again one build at a time, are rows at the edge of the harness's patience, answered at 8 to 24 seconds by both builds where either answers; master answers five of them alone and this four, the fifth 1.1.2.4:343, at 23 seconds on master, which went the same way in two other branches' runs the same night. The figures above were measured with three other corpus runs on the machine.

The suite on the commit measured, 864693e3, passed but for IntegralAnswerCacheTest.AnIntegralWithNoAnswerStillFinishesQuickly, which allows a declined integral thirty seconds and took forty-six with four corpus runs on the machine; its class passes alone in four seconds. The allocation gate passed. The head here, 0f8e76a6, merges master 092b2815; the calculus and corpus tests pass on it and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura