Tips on integral of exponential and error function

Question

In my work I've come across an integral on the form $$ \int\limits_{x_0}^\infty \exp\left[-a x^2 \right] \text{erf} \left[b x + c \right] \ \mathrm{d}x \ , \qquad a > 0, \quad b, c \in \mathbb{R} $$ which I cannot for the life of me figure out. Neither Mathematica nor Maple could help me find a closed-form solution with the given limits. I've found solutions for $x_0 = 0$ and $x_0 = -\infty$, but no general solution for $x_0 \in \mathbb{R}$. I suspect there's no analytic solution, but I thought I'd ask here as a last resort.

With variable substitution and integration by parts, I end up juggling between different versions of $$ \int\limits_{x_0}^\infty x^{2m} \exp\left[-x^2 \right] \text{erf} \left[\beta x + \gamma \right] \ \mathrm{d}x \ , \qquad m \in \mathbb{Z} $$ but I feel no closer to the solution. I've also tried the trick of differentation outlined here, but I end up with the same problem.

Korotkov's Integrals Related to the Error Function gives solutions for integrands with a factor of $x^{2m + 1}$, which leads me to suspect that there isn't any solution for $x^{2m}$.

Any ideas or advice would be greatly appreciated!

This is the $m$th derivative wrt $a$ of $\int e^{-ax^2}\operatorname{erf}(\beta x+\gamma)$. That integral evaluates to a sum of OwenT and trig functions, as done here — Sal, Oct 11 '22 at 08:31
Does this answer your question? How to solve integral involving normal distribution and error function and x — Andrew D. Hwang, Oct 11 '22 at 13:16
@Sal That's it! Thank you so much, the response you linked is exactly what I was looking for. I didn't know about Owen's T-function, but I see it's used to describe situations related to the one that gave me the integral above. If you post an answer, I'll mark it as accepted. — asbjoree, Oct 11 '22 at 19:15

score 2 · Accepted Answer · answered Oct 12 '22 at 09:34

Consider the related integral $$ G(a)=\int\limits_{x_0}^\infty dx \ e^{-ax^2}\operatorname{erf}(\alpha x+\gamma) $$

So that your integral is $\left(-\frac{d}{da}\right)^m G(1)$. It turns out $G(a)$ evaluates to a sum of trig and OwenT functions, as done here and used here to evaluate a similar integral.

Tips on integral of exponential and error function

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