A square of a special function of a shifted argument is asked term by term

[Rafael-SOWNet]

#1639 asked the remainder of a special function's square term by term only for an argument b x. For a + b x, the remainder divided by the linear, and its terms were the exponentials a hyperbolic function is written as, each a harder question than the whole. With the constant of integration matched to the linear (#1642), the linear cancels and the terms are the case without the offset. So the asking is now offered for any linear argument, with one exception.

The exception is an error function beside anything but itself. Its derivative is a Gaussian of the shifted argument rather than a quotient by it, so a polynomial beside it stays in every term. Asking them took the decline of (c + d x) erf(a + b x)^2 from six seconds to twenty-four, and answered nothing.

Measured

Input	Was (2.5.0)	Now
"Shi(a+b*x)^2".Integrate("x")	Shi * (a + b * x) ^ 3 / 3 / b + C, with Shi a variable	an antiderivative in Shi(a + b x), Ei(2 (a + b x)) and Ei(-2 (a + b x))
"x^2*Ei(a+b*x)^2".Integrate("x")	a ^ 2 * Ei * x ^ 3 / 3 + a * b * 2 * Ei * x ^ 4 / 4 + b ^ 2 * Ei * x ^ 5 / 5 + C, with Ei a variable	an antiderivative in Ei(a + b x) and Ei(2 (a + b x))
"erf(a+b*x)^2".Integrate("x")	UnrecognizedFunctionParseException: there is no function erf	an antiderivative in erf

• Rubi, master at 16203aa6 and this change on it, run side by side, 0 wrong everywhere:
  • Family 8: 380 → 394 of 420, 0 timeout. The fourteen:
    • erf, erfc and erfi of a + b x, squared;
    • Si and Ci of a + b x squared, beside x and beside x^2;
    • Ei, Shi and Chi of a + b x squared, beside x^2;
    • Shi(a + b x)^2 and Si(a + b x)^2;
    • x^2 Ci(a + b x) cos(a + b x) and x^2 cos(a + b x) Si(a + b x).
  • Family 2: 544 of 650 on both.
  • The independent suites: 1756 of 1814 on both.
  • Families 1 and 3–7 at five a file: 800 of 897 on both.
• Run alone, the fourteen are 0 of 14 on master and 14 of 14 here.
• With the error functions not excepted, measured the same way, family 8 was the same 394 with six more timeouts: the error functions of a + b x squared beside c + d x or its square. Families 0–7 moved nowhere.
• The unit tests pass: 14,265, none failed (net10.0). The native AOT publish of the C++ wrapper builds with no trim or AOT warnings.
• The performance gate passes on ec42a120. Allocation matches the baseline on all 19 gated benchmarks.

SpecialFunctionsByPartsTest has four more rows with a shifted argument, each differentiated back with its parameters pinned.

Part of #1501.

🤖 Generated with Claude Code