TITLE: How to evaluate the integral $\int_{-1}^1 e^{ax^2+bx+c\sqrt{1-x^2}}dx$
QUESTION [5 upvotes]: Can anyone show if the following integral can be evaluated in closed form? 
\begin{equation}
\int_{-1}^1 e^{ax^2+bx+c\sqrt{1-x^2}}dx
\end{equation}
The variable $x$ can be replaced by $\cos{\theta}$, with corresponding change of the interval of integration.

REPLY [7 votes]: There can't be a closed form for that integral. If there was a closed form, $I(a,b,c)$, then set $b=c=0$ and $a=-1$ to get a closed form for $\int_{-1}^{1} e^{-x^2}dx$, which has been shown to not exist.