I am trying to resolve this integral, which comes out of considering a compound distribution of normal variables:
$$ \int_{-\infty}^{\infty} \frac{1}{\sigma_{\sigma} \sqrt{2 \pi}} \frac{1}{\sqrt{\hat{\sigma}^2 + \sigma_{\mu}^2}\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2(\hat{\sigma}^2 + \sigma_{\mu}^2)}\right) \exp\left(-\frac{(\hat{\sigma}-\sigma)^2}{2\sigma_{\sigma}^2}\right) d \hat{\sigma} $$ I would love help finding an analytic solution, or even just approximations. You can assume all parameters are greater than $0$, and real.