SCHWEITZER 2004 Problem 10

Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$ , let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$ . For $p>1$ , prove that $$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$