I've calculated and seen expectation of $\sin X$ when $X \sim N(\mu, \sigma^2)$. Popular method seems to be using complex variables to calculate $\int \sin x \cdot \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} = Im(\int e^{ix} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}) = e^{-\frac{\sigma^2}{2}}\sin \mu$.
However, it was impossible for me, until now, to imitate such approach in calculating expectation of $|\sin X|$. I've failed to found complex variable representation of $|\sin x|$, and also have not figured out any other ways to calculate $\int |\sin x| \cdot \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$. I tried splitting it up to infinite sum making it $ \sum_{k=-\infty}^{k=\infty} \int_0^\pi \sin x \cdot \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-k\pi-\mu)^2}{2\sigma^2}} $ but this integration in bounded interval is tough, too.
Strict equation in $\mu, \sigma$ will be the best, but any good upper or lower bounds is appreciated, too(but hopefully not something like 1 or 0). I am quite stuck with no idea where to go. Thanks all in advance.