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A root of one in the sine's symbolic quadratic is taken as the square it makes - #1596
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… it makes csc(x)^2/(a - a sin(x)^2) and sin(x)^6/(a - a sin(x)^2) were answered with NaN wherever they were evaluated. The rule for a rational function of the sine over a symbolic quadratic in it splits over the quadratic's two roots and integrates each 1/(sin(x) - r) as -2 atan((r tan(x/2) - 1)/sqrt(r^2 - 1))/sqrt(r^2 - 1). Its documentation excludes r = 1 and r = -1, but nothing checked for them, and a - a s^2 has exactly those roots for every a. Where r^2 - 1 vanishes identically, the quadratic under the half-angle is a square and the piece is rational in t = tan(x/2): 2/(r t - 1) for the sine, and for the cosine (1 + r)/(2t) + (1 - r) t/2, which is 1/t at 1 and t at -1. Measured with work/intbench against dd72825, 3 s a problem: Rubi's 102 rows with a - a sin^2, a - a cos^2 or a - b sin^2 below the bar, 88 -> 102 solved, the 14 wrong -- all of 4.1.7's rows 64 to 77 -- none wrong now. Closes #1588. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Closes #1588: a rational function of the sine or the cosine over a symbolic quadratic whose roots are ±1 is integrated, where it was answered with NaN.
SolveARationalFunctionOfASineOverASymbolicQuadraticsplits over the two rootsrof the quadratic in the function, and integrates each1/(sin(x) - r)as-2 atan((r tan(x/2) - 1)/sqrt(r^2 - 1))/sqrt(r^2 - 1). Its documentation excludesr = ±1, but nothing checked for them.a - a sin(x)^2has exactly those roots for everya, so the answer divided by zero wherever it was evaluated.Where
r^2 - 1vanishes identically, the quadratic under the half-angle is a square, and the piece is rational int = tan(x/2):2/(r t - 1);(1 + r)/(2t) + (1 - r) t/2, which is1/tat 1 andtat -1.csc(x)^2/(a - a*sin(x)^2)integral(...)-cot(x)/aand two terms rational intan(x/2)sin(x)^6/(a - a*sin(x)^2)integral(...)1/(a - a*cos(x)^2)integral(...)tan(x/2)cos(x)^4/(a - a*cos(x)^2)integral(...)SineOverASymbolicQuadraticTest.OverRootsAtOnedifferentiates four such integrands back ata = 1.3and ata = -0.7.Measured with
work/intbenchagainst master atdd728251, the commit this is cut from, 3 s a problem:a - a sin(y)^2,a - a cos(y)^2ora - b sin(y)^2below the bar (4.1.7 and 4.2.7): 88 → 102 solved, 14 wrong → 0. The fourteen are all of 4.1.7's rows 64 to 77, powers of the sine and the cosecant overa - a sin(c + d x)^2, which master answered with NaN.dd728251: re-run on each branch alone, none of the twelve moves is this branch's, and none is a loss.SineOverASymbolicQuadraticTestpasses (15), and every target builds.🤖 Generated with Claude Code