A special function beside an elementary factor is integrated by parts, and so is its square

[Rafael-SOWNet]

x Si(b x) sin(b x) was left unevaluated where Si(b x) sin(b x) was not, and so was every square of a special function beside a power of x. Two steps of integration by parts were missing, and both are scoped to a special function of a linear argument.

• The factor beside the special function, by its polynomial's parts. By parts, x Si(b x) sin(b x) is Si(b x) against x sin(b x). The integral of that factor was taken with parts off, and x sin(b x) needs them. Beside a special function of a linear argument, a polynomial times something else is now integrated by the polynomial's parts, which end in as many steps as its degree. What is left, the special function's derivative times polynomials and elementary functions, is given parts in turn.
• The remainder of a square, term by term. A square of a special function of b x is two rounds of parts. The remainder of the first comes back as one product over a sum that no rule reads whole: (b x Ei(b x) - e^(b x)) e^(b x)/(b x) for Ei(b x)^2. Its terms, e^(b x) Ei(b x) and e^(2 b x)/(b x), are each answered when asked. So where the remainder after a special function is not answered, its terms are asked one at a time, the ordinary way first and then as questions of their own. That is bounded three ways:
  • At the top only, where such asking reaches.
  • Once. A term asked so is at the top itself, and asking inside it lifted the search again with the depth reset each time. Together with a rule that wrote a + b x as t, x Ei(a + b x)^2 grew a test host to twelve gigabytes; that rule is not in this PR. A decline the once-only flag causes is not cached (Integration.DeclinedForItsScope), as a decline that ran out of depth is not.
  • For a multiple of x only. For a + b x, the remainder divides by the linear, and its terms are the exponentials a hyperbolic function is written as. Asking them took the decline of x Shi(a + b x)^2 from a second to past twenty.

Measured

Input	Was (2.5.0)	Now
"x*Si(b*x)*sin(b*x)".Integrate("x")	a polynomial in a variable Si	an antiderivative in Si(b x), Ci(2 b x) and ln(x)
"Ei(b*x)^2".Integrate("x")	Ei * (b * x) ^ 3 / 3 / b + C, with Ei a variable	an antiderivative in Ei(b x) and Ei(2 b x)
"x*erf(b*x)^2".Integrate("x")	UnrecognizedFunctionParseException: there is no function erf	an antiderivative in erf(b x)

• Rubi's family 8, Special functions, first tranche: the error functions and the exponential, logarithmic, trigonometric and hyperbolic integrals #1501's tranche of 420: 296 → 367 solved, 0 wrong, 0 timeout, against master with The special functions are substitutions, and powers of one base gather across the bar #1638.
  • The 71 gains are 0 of 71 on master and 71 of 71 here, run alone.
  • The 53 still declined take 103 s against 117 s together, with a median ratio of 1.00. The slowest change is x^2 Ci(a + b x) cos(a + b x) and its sine sibling, 0.25 s to 2 s.
• Nothing else moves. The independent suites are 1756 of 1814 on both, and families 1–7 at five problems a file are 813 of 912 on both. The problems declined in both take what they did (11.9 s against 11.5 s, 98.1 s against 98.9 s). 0 wrong everywhere.
• The unit tests pass: 14,247, none failed (net10.0). The native AOT publish of the C++ wrapper builds with no trim or AOT warnings.
• The performance gate passes on ee5cb79e, whose tree is this head's but for the BREAKING-CHANGES entry. Allocation matches the baseline on all 19 gated benchmarks.

SpecialFunctionsByPartsTest has six rows beside a power and the elementary factor of the derivative, and eleven squares, each differentiated back with its parameters pinned.

Not in this PR: the squares of a special function of a + b x beside a power of x, about twenty of the 53 left. By parts leaves them over the linear, and they want the polynomial divided by it before the terms are asked. Also F(d (a + b log(c x^n))) (16), and the Gaussian-weighted erf(b x)/(e^(b^2 x^2) x^k) (8).

Part of #1501.

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