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.

I came across this integral because I wanted to derive an equation that is the Fourier Transform of a skewed Gaussian function.

$$ \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \left( 1 + \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\right) \right) $$

the first term is a regular Gaussian function. The second term is a complicated one because it has the error function also. Is there a way to approximate the skewed Gaussian function so that it could have a Fourier Transform based on the parameters $\alpha, \mu$ and $\sigma$ ?