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.
