Is there any closed form, asymptotics, and/or approximations for the following integral:


$$f(c) := \frac{1}{\sqrt{2\sigma^2\pi}}\int e^{-x^2/2\sigma^2}\frac{x^2}{1-cx^2}dx,$$ where $\sigma^2$ is real valued and $c$ is complex? I am integrating over $\mathbb{R}$.


Even in the case where $c$ is real would be interesting! Pointers to relevant references are appreciated!



Evaluation of Gaussian density integral
Obviously you need $\sigma^2 > 0$ (and I'll assume $\sigma > 0$), and $c$ should not be a positive real to avoid having singularities at $x = \pm \sqrt{1/c}$. According to Maple, the result is


$$ -{\frac {\sqrt {\sigma}}{c}}- \frac{\sqrt{-\pi c}}{ c^2 \sqrt{2\sigma}}
\exp\left(-\frac{1}{2c\sigma^2}\right) + \frac{\sqrt{\pi}}{\sqrt{2\sigma} c^{3/2}} \exp\left(-\frac{1}{2c\sigma^2}\right) \text{erfi}\left(\frac{1}{\sigma \sqrt{2c}}\right)$$


This appears to check out numerically.


