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An antiderivative built through an even root says where the root is real - #1431
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The linear-radical substitution u = (a x + b)^(1/q) and the general substitution for an even root both take a power of u out of a radical on the strength of u being a non-negative real, which it is only where the radical is. Beyond the radicand's zero u is imaginary and the integrand can still be real: Rubi's x sqrt(c - a c x)/e^(3 atanh(a x)) above a x = 1 is the product of two imaginary factors, and the answer built for a real u was 14% off its derivative there and shipped as an answer (family 7, one wrong after #1430 made the row reachable). An answer that used the step now carries provided a x + b >= 0: NaN beyond it, which is no claim, and unchanged where it holds. An answer the step did not touch is as it was. #718 Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Found by the Rubi family-7 run on #1430's merge commit:
x*sqrt(c-a*c*x)/e^(3*atanh(a*x))(7.3.6) came back wrong — d/dx 14% off the integrand atx = 0.59,a = 3.1,c = 0.7. Not a defect of the new rule:e^(3 atanh(a x))now folds to(2/(1 - a x) - 1)^(3/2)(exact), which made the row reachable for the linear-radical substitutionu = sqrt(c - a c x), and that route was already wrong on the same shape (sqrt(c-a*c*x)*(2/(1-a*x)-1)^(-3/2)on master, WRONG 2/6).The cause. For an even
qthe substitution takes a power ofuout of the radicals —(2c/u^2 - 1)^(-3/2)asu^3/(2c - u^2)^(3/2)— on the strength ofubeing a non-negative real (FactorANonnegativeVariableOutOfRadicals), which it is only where the radical is real. Abovea x = 1the root is imaginary, the identity does not hold, and the integrand is still real there:sqrt(negative) · (negative)^(-3/2)isi · i · (…). So the answer built for a realuis not the antiderivative on that region, and nothing said so. The general substitution's even-root branch (SolveBySubstitution, "an even root is not negative") made the same assumption.The fix. An answer that used the step carries the condition it used:
provided a x + b >= 0(the linear-radical route, unless the caller knowsunon-negative, asu = e^(k x)is),provided radicand >= 0(the general route). Beyond it the answer isNaN— no claim — and inside it is unchanged. An answer the step did not touch (x/sqrt(1 + x),1/(sqrt(x) - x^(-1/3))) is as it was.1/sqrt(x + x^(3/2))now reads2 * sqrt(1 + sqrt(x)) / (1/2) + C provided x >= 0.LinearRadicalSubstitutionIntegralTest.TheEvenRootStepSaysWhereItHoldspins both: the condition on the answer, the derivative matching inside it at three points, the integrand real atx = 0.59and the answer's derivativeNaNthere. Full suite 12007 passed, 0 failed. BREAKING-CHANGES.md entry measured on a v2.5.0 build (both rows were unevaluated there).The harness side is in the workspace:
intbenchnow skips a sample point where a condition the answer states (anywhere in its tree, sinceCis added outside theprovided) is decidedly False, instead of stripping the condition and comparing — an answer claims nothing there. On that harness: 1774 suite 1706/1774, 0 wrong, 0 timeout (unchanged), family 7 153/270, 0 wrong (152 with the wrong row before), family 6 315/417, 0 wrong (unchanged).Part of #718.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura