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…r across the bar (#1638) None of the nine special functions was a candidate for the substitution u = g(x), so an integrand that is a power of one beside its derivative had no route: e^(c - b^2 x^2) erf(b x)^n, Ei(b x) e^(b x)/x and Si(b x) sin(b x)/x were left unevaluated. Each is a candidate now, as the logarithm is, and its elementary derivative is the differential: e^(c - b^2 x^2) erf(b x) is sqrt(pi) e^c/(2b) u du under u = erf(b x). A constant in the exponent still stopped it: divided by du/dx, e^(c - b^2 x^2) over the e^(-(b x)^2) of the derivative is e^c, and the quotient is simplified only a level, so the x in it was left and the candidate refused, where e^(-b^2 x^2) erf(b x) was answered. The divisor's factors are spread to the power -1 now, the powers of one base gathered, and the result simplified a level, which is where e^(c - (b x)^2 + (b x)^2) becomes e^c. Both only where they can answer. A special function is offered only where its derivative can be the differential: an exponential of a quadratic for the error functions, a logarithm below the bar for li, and something in x below the bar for the others, whose derivatives divide by their argument. Offered beside anything, x cosh(a + b x) Shi(a + b x) went from a second to ten, simplifying quotients of exponentials that were never going to lose their x. And the powers are gathered only for a special-function candidate: done for every candidate that left an x, the exponentials of 1/((c + d x)^3 (a + a tanh(e + f x))) were gathered and simplified for each one, and seven declines of a third of a second became timeouts. The nine are listed once, in IsASpecialFunction, where the solver listed them three times. Rubi's family 8, #1501's tranche of 420: 258 -> 296 solved, 0 wrong, against master at ad05a79. Nothing without a special function moves. The independent suites are 1753 of 1814 on both; families 1 to 7 at five problems a file are 809 and 811 of 912 run side by side, and the two that moved, 3.2.1 #12 and 6.7.1 #180, are answered alone by both, in 18 and 22 s: master's timeouts there were the load. 0 wrong everywhere. The unit tests pass on master with #1637, 14,230, and the performance gate passes on fd1a4416, which is this change on ad05a79: allocation is what the baseline says on all 19 gated benchmarks. SpecialFunctionSubstitutionTest has nine rows from Rubi's 8.1, 8.3 and 8.4, each differentiated back with its parameters pinned. Part of #1501. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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