A special function of a logarithm of a monomial is integrated, through Ei of a complex argument

[Rafael-SOWNet]

(e x)^m Si(d (a + b ln(c x^n))) is by parts against (e x)^m, and what is left is a power of x times sin(d (a + b ln(c x^n)))/(a + b ln(c x^n)). Two rules were missing on the way, and both are here.

• A function of one logarithm of a monomial. x^m G(ln(c x^n)) is integrated under t = ln(c x^n). dx is x dt/n, and x^(m + 1) is K e^((m + 1) t/n) with K = x^(m + 1) (c x^n)^(-(m + 1)/n), whose derivative is 0 wherever it is defined. So the answer is K/n times the integral of e^((m + 1) t/n) G(t) at t = ln(c x^n). That is an antiderivative wherever the integrand is real. For an even n that includes negative x, where ln(c x^n) is not ln(c) + n ln(x) and an answer through ln(x) would be off by a branch. A power of a monomial, (e x)^m, is x^m times a factor of the same kind. Rubi's answers to 8.3–8.5 are written in this K. The integral in t is asked as one quotient, as the logarithm substitution asks its own: as a product, (d x)^m x/ln(b x) handed over e^((m + 2) t) t^(-1), which ran away in integration by parts (Integrating e^(2x) x^(-1) runs out of memory: integration by parts reads x^(-1) as a polynomial #1646, fixed by A negative power of the variable is not a polynomial to integration by parts #1647). As a quotient it is Ei((m + 2) t) at once, and (d x)^m li(b x) is answered too, since by parts leaves (d x)^m x/ln(b x) of it.
• An exponential beside sines and cosines, over linears. Each sine and cosine is written as exponentials and the product multiplied out. Every term is then an exponential with a complex rate over the linears, which the exponential rules answer with Ei of a complex argument. e^(2x) sin(x)/x is (Ei((2 + i) x) - Ei((2 - i) x))/(2i), two conjugate terms whose sum is real. This applies only with an exponential beside the sine or cosine: sin(x)/x alone stays Si(x).

The evaluator already reads Ei, Si and erf off the real line, so these answers are checked like any other. Ei(1 + 2i) evaluates to 1.0421677… + 3.7015014… i.

Measured

Input	Was (2.5.0)	Now
"e^(2*x)*sin(x)/x".Integrate("x")	integral(e ^ (2 * x) * sin(x) / x, x)	(-1/2 * i) * Ei((2 + i) * x) + 1/2 * i * Ei((2 - i) * x) + C
"x*sin(ln(x))/ln(x)".Integrate("x")	integral(x * sin(ln(x)) / ln(x), x)	(-1/2 * i) * Ei((2 + i) * ln(x)) + 1/2 * i * Ei((2 - i) * ln(x)) + C
"cos(a+b*ln(c*x^n))^2".Integrate("x")	integral(cos(a + b * ln(c * x ^ n)) ^ 2, x)	an antiderivative in sin and cos of 2 (a + b ln(c x^n)), beside x (c x^n)^(-1/n)

• Rubi, master at 492afc23 and this change on it, run side by side, 0 wrong everywhere:
  • Family 8: 395 → 411 of 420:
    • all fifteen of F(d (a + b ln(c x^n))): Ei, Shi and Chi beside (e x)^m, and Si and Ci alone, beside x, x^2 and (e x)^m, and over x^2 and x^3;
    • (d x)^m li(b x), Rubi's 8.3 row 269.
  • Family 4 at ten a file: 611 → 612, the one being cos(a + b ln(c x^n))^2.
  • Family 2: 544 of 650 on both.
  • Family 3 at twenty a file: 171 of 180 on both.
  • The independent suites: 1756 of 1814 on both.
  • Families 1, 5, 6 and 7 at five a file: 437 of 499 on both.
• Run alone, the seventeen are 0 of 17 on master and 17 of 17 here.
• The unit tests pass on 6a4b09c1, this change on 492afc23: 14,275, none failed (net10.0), with the GC heap held to 4 GB. Rebased onto master with A negative power of the variable is not a polynomial to integration by parts #1647, the integration test classes pass (199).
• The native AOT publish of the C++ wrapper builds with no trim or AOT warnings.
• The performance gate passes on 6a4b09c1. Allocation matches the baseline on all 19 gated benchmarks.

ExponentialIntegralIntegrationTest has three rows of an exponential beside a sine or cosine over a linear. It also has four of a function of a logarithm of a monomial, checked at negative x as well with n = 2, and (k x)^m x/ln(b x) past 1. SpecialFunctionsByPartsTest checks (d x)^m li(b x) at positive points, where li(b x) is real.

Part of #1501.

🤖 Generated with Claude Code

[Rafael-SOWNet]

On hold, and back to draft: (d x)^m x/ln(b x) runs out of memory with this change. The new rule hands the integrator e^((m + 2) t) t^(-1), and master runs away on that spelling too (#1646). I will push the integrand built as one quotient, re-measured, and mark this ready again.

[Rafael-SOWNet]

Ready again. The integral in t is now asked as one quotient (b6f0d99), so (d x)^m x/ln(b x) is Ei((m + 2) ln(b x)) at once, and (d x)^m li(b x) is answered as well. Re-measured against master at 492afc2: family 8 395 → 411 of 420, family 4 at ten a file +1, the rest unchanged, 0 wrong. The suite and the gate pass. Rebased onto master with #1647.