Suppose $X\sim \mathcal{N}(0,\sigma^2)$, find the expectation $\mathbb{E}\left[\frac{1}{(1+X^2)^a}\right]$ where $a$ is a fixed positive real number.


Is there an explicit formula for the above expectation?



Expected value of a function of normal random variable
According to Mathematica, the expectation is
$$\frac{\Gamma \left(a-\frac{1}{2}\right) \, \_1F\_1\left(\frac{1}{2};\frac{3}{2}-a;\frac{1}{2 \sigma^2}\right)}{\sqrt{2} \sigma
 \Gamma (a)}+\frac{2^{-a} \sigma^{-2 a} \Gamma \left(\frac{1}{2}-a\right) \, \_1F\_1\left(a;a+\frac{1}{2};\frac{1}{2
 \sigma^2}\right)}{\sqrt{\pi }}.$$


