Evaluating an integral involving a Gaussian function I am stuck on the following integral:
$$I(a_1,\ldots,a_n)=\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\exp\left[-\frac{1}{2}\sum_{i=1}^n x_i^2+\frac{\sum_{i=1}^na_ix_i}{\sqrt{\sum_{i=1}^n x_i^2}}\right]dx_1\cdots dx_n$$
where $a_1,\ldots,a_n$ are arbitrary finite real numbers.  If the integral cannot be evaluated in closed-form, I am wondering if $I(a_1,\ldots,a_n)$ is a monotonic function of $\sum_{i=1}^n a_i^2$.  
This is related to this question on stats.SE.
 A: Note that $I(a)=E[\exp(a\cdot X/\|X\|)]$ where $X$ is standard normal in $n$ dimensions. The invariance by rotation of the distribution of $X$ indicates that $X/\|X\|$ is uniform on the unit sphere $S^{n-1}$ hence $I(a)=E[\exp(a\cdot U)]$ where $U$ is uniformly distributed on $S^{n-1}$. 
The distribution of $a\cdot U$ does not change when $a$ is replaced by another vector with the same norm hence $I(a)=E[\exp(\|a\|U_1)]$, where $U_1$ denotes the first coordinate of $U$. The distribution of $U_1$ is characterized by the fact that $U_1=\cos(\Theta)$ where the density of $\Theta$ is proportional to $\sin^{n-2}$ on $[0,\pi]$. Hence, $I(a)=F_n(\|a\|)/F_n(0)$ with
$$
F_n(\alpha)=\int_0^{\pi}\mathrm e^{\alpha\cos(t)}\sin^{n-2}(t)\mathrm dt=\int_{-1}^1\mathrm e^{\alpha x}(1-x^2)^{(n-3)/2}\mathrm dx,
$$
which can be expanded as a series in $\alpha^2$ and probably identified as a special function, since
$$
F_n(\alpha)\propto\sum_{k\geqslant0}\frac{\Gamma(k+n/2)\alpha^{2k}}{\Gamma(2k+1)\Gamma(k+1/2)}.
$$
In particular, there exists some positive coefficients $(i_k)_k$ such that
$$
I(a)=\sum_{k\geqslant0}i_k\,\|a\|^{2k},
$$
hence $a\mapsto I(a)$ is increasing with respect to $\|a\|$ from $I(0)=1$ to $\lim\limits_{\|a\|\to\infty}I(a)=+\infty$..
