I have tried to calculate the following integral in terms of $\sigma^2$ :
$I=\int_{x=-\infty}^{+\infty}\frac{1}{\sqrt{1+x^2}}e^{-x^2/{2\sigma^2}} dx$
However by using conventional methods such as changing variables I canmot reach to a closed-form solution even with Q-function.
I appreciate any help in determining the solution of this integral.
My unsuccessfull tries are:
1- change of variable $u^2 := \frac{1}{\sqrt{1+x^2}}$ is resulted a nother more complicated gaussian Integral of this form.
2-representing in the polar system also have the same problem as above.
3-using enter link description here does not give answer.
4-checking enter link description here
5-integral by parts.