I want to get a closed/ semi-closed form of the integral given below.

$$ \int_{-\infty}^{+\infty} \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\right) \exp{(i x \tau)} dx$$

Where $\text{erf}$ is the error function and $i = \sqrt{-1}$. As $\text{erf}$ itself is an integral, this is a double integral already.