I wonder if there is an analytical solution to
$$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$
where $a, b>0$.
I know, of course, that the antiderivative of the fraction is a version of $\arctan$, and thus tried partial integration, but unfortunately this won't help. (I plugged it into Wolfram Alpha but it exceeded the computation time...) I very much appreciate your effort!
EDIT: Maybe it is appropriate to rephrase the question a bit: I am searching for a closed-form or analytically tractable solution for the expected value of $\frac{1}{aX^2+b}$ where $X\sim N(\mu,\sigma)$ and $a,b>0$.
Depends what you mean by that. If you are willing to accept imaginary error functions as being “analytical”, then the answer is yes. Alternatively, one might try and expand $\dfrac1{(x+a)^2+b}$ into a binomial series, and then reverse the order of summation and integration in order to rewrite the integral in terms of hypergeometric functions, but I'm afraid this might not work unless $b=a^2$.