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

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.

• You should at least show your effort. By the way, I think there should be a closed form in terms of zeta function. – Szeto Jul 8 '18 at 5:08
• @Szeto. Just out of curiosity : how did you think about a possible zeta function ? I thought about some nasty beta function, but, for sure and as usual, I could be totally wrong. Cheers. – Claude Leibovici Jul 8 '18 at 6:16
• @ClaudeLeibovici I just did math in my mind for 30 seconds before writing the comment. Don’t take anything seriously. – Szeto Jul 8 '18 at 7:55