# Gaussian integral over logarithm of shifted error function

Suppose we have the following integral:

$$\int^{\infty}_{-\infty} \frac{dz}{\sqrt{2\pi}}e^{\frac{-z^2}{2} } \log\left( \text{erf} \, a (z-b) +1 \right), \ \ \ \ a,b \in \mathbb{R}$$ Does a closed-form solution of this integral exist? If it does not exist, does it exist for $b=0$?

What I tried so far

Even numerical evaluation of the problem for given $a,b$ is difficult, since the error-function becomes -1 for large negative z causing the logarithm to diverge.

On Wikipedia I found the following solution for a related problem: $$\int^{\infty}_{-\infty} \frac{dz}{\sqrt{2\pi d^2}}\text{exp}\left[{\frac{-(z+c)^2}{2 d^2} }\right] \text{erf}(az+b)=\text{erf}\left[\frac{b-ac}{\sqrt{1+2a^2d^2}} \right],$$ which makes me suggest that something similar might exist for the stated problem.

• Googling “Ng Geller Erf Nist” will get you two tables of erf-related integrals. The integral you want may not exist, but you already found the one that seems to be helpful most often, and you can see if any of their other results help. Apr 14 '18 at 19:48
• even for $b=0$, $a=1$ a closed-form solution is not forthcoming Apr 14 '18 at 20:14