Expected value of log of 1+ a squared Gaussian random variable

Question

If $X$ is standard normal, what is

$$\mathbb{E}\log(1+X^2).$$

I see that $X^2\sim\mathrm{Gamma}(\frac 12,2),$ but is there a simple formula for the above (perhaps in terms of the polygamma functions)?

Answer 1

We have to compute: $$\begin{eqnarray*}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\log(1+x^2)\,e^{-x^2/2}dx&=&\frac{1}{\sqrt{2\pi}}\lim_{s\to 0^+}\frac{d}{ds}\int_{-\infty}^{+\infty}(1+x^2)^s\,e^{-x^2/2}\,dx=\\ &=&\frac{1}{\sqrt{2\pi}}\lim_{s\to 0^+}\frac{d}{ds}U\left(\frac{1}{2},\frac{3}{2}+s,\frac{1}{2}\right)\end{eqnarray*}$$ where $U$ is the confluent hypergeometric function.