How to get the closed form of integration of sigmoid function multiply normal distribution density function

Question

As the title describes, I want to get the closed form of the following equation $$ f(\sigma,\mu) = \int_{-\infty}^{\infty} \frac{1}{1+e^{-(\beta X + \alpha)}}\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(X-\mu)^2}{2 \sigma^2}} dX $$

$\beta$ and $\alpha$ are known, only $\mu$ and $\sigma$ are variables.