An exponential of a quadratic in a logarithm beside a power of x is integrated onto the Gaussian

[Rafael-SOWNet]

Part of #1501.

x^p G^(Q(L)), with L = ln(c x^r) and Q a quadratic with a square term, is now integrated. Under t = L, with L' = s/x, dx = x dt/s, and x^(p + 1) e^(-(p + 1) L/s) is constant. So the integral is that constant over s times the integral of e^(Q(t) ln G + (p + 1) t/s): the Gaussian with a linear term, which the table answers (#1512). This is #1518's substitution with t = L in place of t = sqrt(F), and the constant is written as #1518 writes it, and as Rubi does.

A polynomial beside it is summed a power at a time. The logarithm of a power of a linear is the same question under u = d + e x. A constant multiple of that linear, as Rubi's (d g + e g x)^m writes one, is read as g u. It is a power only in u, so the logarithm is found, and the substitution made, before any factor is read.

integral	master	this PR
e^(ln(x)^2), x e^(ln(x)^2), 2^(ln(x)^2 + ln(x))	unevaluated	answered, with erfi
x^m F^(f (a + b ln(c x^n))^2)	unevaluated	answered
F^(f (a + b ln(c (d + e x)^n))^2) (d g + e g x)^m, m symbolic, 2, 1, -2, -3	unevaluated	answered
the same with (g + h x)^3, ^2, ^1	unevaluated	answered
the same with F^(f (a + b ln(c (d + e x)^n)^2)), the square inside	unevaluated	answered

Measured

Measured with the change on 22021b1c, before the rebase onto #1520. After the rebase, the library builds for every target, and the logarithm and Gaussian test classes pass (86).

• Rubi's problems whose answer uses erf, erfc or erfi: 811 across families 0 to 7, at a 3 s budget. This PR solves 614, where master (22021b1c) solves 593, with 0 wrong and nothing lost. Both have the same 14 timeouts, and the total time goes from 574 s to 568 s. The 21 gained are all in 2.3, and every one of its 23 exponentials of a quadratic in a logarithm is now solved; the two below d g + e g x alone were solved already.
• The textbook suites (family 0) are unchanged: 1723/1778, 0 wrong, 0 timeouts.
• The unit suite passes on net10.0: 12956 passed, 14 skipped, none failed, of 12970.
• The allocation gate passes on all 19 gated benchmarks, with no allocation moved.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura