TITLE: $\int_{-\infty}^{+\infty}\frac1{1+x^2}\left(\frac{\mathrm d^n}{\mathrm dx^n}e^{-x^2}\right)\mathrm dx$ Evaluate
QUESTION [18 upvotes]: How can I solve this problem?
$$
\int_{-\infty}^{+\infty}\frac1{1+x^2}\left(\frac{\mathrm d^n}{\mathrm dx^n}e^{-x^2}\right)\mathrm dx
$$

REPLY [7 votes]: $$
\int_{-\infty}^{+\infty}\frac1{1+x^2}\left(\frac{\mathrm d^n}{\mathrm dx^n}e^{-x^2}\right)\mathrm dx = \,U\left(1+\frac{n}{2},\frac{3}{2},1\right)\cos\left(\frac{\pi\,n}{2}\right)\,n!\,\sqrt{\pi},
$$
where $U(a,b,z)$ is the confluent hypergeometric function of the second kind. 
The proof is quite long. I started with applying the Fourier transform to $e^{-x^2}$ so that the derivative $\frac{\mathrm d^n}{\mathrm dx^n}$turns into a multiplication by $(-i\, t)^n$. The inverse transform gives a general form for n-th derivative in terms of hypergeometric functions.