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A power of x times the square of erf of a shifted argument is not integrated #1945
Description
Part of #718.
Rubi 8.1 integrates (c + d x)^m erf(a + b x)^2. On master (3d5d146), as of 2.6.0:
| integrand | result |
|---|---|
erf(a + b x)^2 |
answered, 0.9 s |
x erf(b x)^2, x^2 erf(b x)^2 |
answered |
x erf(a + b x)^2 |
integral(…) after 2.4 s |
x^2 erf(a + b x)^2 |
integral(…) after 2.4 s |
x erf(b x) erf(a + b x) |
integral(…) after 8 s |
(c + d x)^2 erf(a + b x)^2 |
times out after 20 s |
So the shift a and a power of x together are what is missing. Integrating by parts leaves x^n e^(-(a + b x)^2) erf(a + b x). With a = 0 this reduces through t = x^2. With a ≠ 0 it needs x = (u - a)/b, after which it is a polynomial in u times e^(-u^2) erf(u), which closes in terms of erf(sqrt(2) u) and e^(-2 u^2).
2.5.0 has no erf integration at all, so this is a gap, not a regression.
🤖 Generated with Claude Code
Activity
Integrate a polynomial beside error functions of a shifted linear (#1947
Rafael-SOWNet commentedon Oct 19, 2026
#1947 (merged, bdbfcb4) integrates
(c + d x)^m erf(a + b x)^2, and the erfc and erfi forms, by writing them in u = a + b x. It also answers Rubi 8.1:52, 181 and 310.Still open:
x erf(b x) erf(a + b x). Its two error functions have different arguments, and integrating by parts leavese^(-b^2 x^2) erf(a + b x), which is not elementary.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Rafael-SOWNet commentedon Oct 22, 2026
Re-measured on master
ec0d9633(2.6.0). Every row of this issue's shape is answered and differentiates back to its integrand:erf(a + b x)^2x erf(a + b x)^2x^2 erf(a + b x)^2x^3 erf(a + b x)^2(c + d x)^2 erf(a + b x)^2x erf(b x) erf(a + b x)is still declined, and that is the right answer. Integrating by parts leavese^(-b^2 x^2) erf(a + b x)ande^(-(a + b x)^2) erf(b x). Their sum, with equal weights, is the derivative oferf(b x) erf(a + b x), but the weights are equal only whena = 0. Otherwise the remainder needs Owen's T function, which the library has no node for. Closing as delivered by #1947.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura