The special functions are substitutions, and powers of one base gather across the bar

[Rafael-SOWNet]

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, and these were left unevaluated:

• e^(c - b^2 x^2) erf(b x)^n
• Ei(b x) e^(b x)/x
• Si(b x) sin(b x)/x

Each special function 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. The quotient was 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 now spread to the power -1, 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.
• Offered only where its derivative can be the differential. That means an exponential of a quadratic for the error functions, and a logarithm below the bar for li. For the others, whose derivatives divide by their argument, it means something in x below the bar. 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.
• Gathered only for a special-function candidate. Done for every candidate that left an x, it gathered and simplified the exponentials of 1/((c + d x)^3 (a + a tanh(e + f x))) once for each candidate. Seven declines of a third of a second became timeouts.

The nine functions were listed three times in the solver, so they are listed once now, in IsASpecialFunction.

Measured

Input	Was (2.5.0)	Now
"e^(c - b^2*x^2)*erf(b*x)".Integrate("x")	UnrecognizedFunctionParseException: there is no function erf	e ^ c * pi ^ (1/2) * erf(b * x) ^ 2 / 2 / (2 * b) + C
"e^(b*x)*Ei(b*x)/x".Integrate("x")	Ei * e ^ (b * x) + C, with Ei a variable	Ei(b * x) ^ 2 / 2 + C
"Si(b*x)*sin(b*x)/x".Integrate("x")	Si * -cos(b * x) + C, with Si a variable	Si(b * x) ^ 2 / 2 + C

• Rubi's family 8, Special functions, first tranche: the error functions and the exponential, logarithmic, trigonometric and hyperbolic integrals #1501's tranche of 420: 258 → 296 solved, 0 wrong, against master at ad05a79b.
• Nothing without a special function moves:
  • The independent suites are 1753 of 1814 on both.
  • Families 1–7 at five problems a file are 809 and 811 of 912, run side by side. The two that moved, 3.2.1 Momo branch #12 and 6.7.1 Let's steal some issues from https://github.com/davidedc/Algebrite! >:3 #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 this head, on master with A rational integral over the roots of its denominator #1637: 14,230, none failed (net10.0). The native AOT publish of the C++ wrapper builds with no trim or AOT warnings.
• The performance gate passes on fd1a4416, which is this change on ad05a79b. Allocation matches the baseline 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.

🤖 Generated with Claude Code