I want to calculate the expectation of one random variable $\frac{1}{\sqrt{a^2+x^2}}$, where $x\sim N(0,\sigma^2)$ and $a$ is a constant.
It is straightforward that we can come to the integration of the following $$\int\limits _{-\infty}^{\infty} \exp(-\frac{x^2}{2\sigma^2}) \frac{1}{\sqrt{x^2+a^2}} dx.$$
How can we handle this integration, or how can we approximate its value?