Evaluate the integral $\int_{-1}^1\exp\left(ax+b\sqrt{1-x^2}\right)\,dx$ I wonder if there is a closed form expression for the following integral with respect to $x$:
\begin{equation}
\int_{-1}^1\exp\left(ax+b\sqrt{1-x^2}\right)\,dx
\end{equation}
where $a,b$ are some constant. Thanks so much! Any help would be greatly appreciated!

$$\int_{-1}^1 \exp \left(a x+b \sqrt{1-x^2}\right) \, dx=\int_{-1}^1 \exp (a x) \exp \left(b \sqrt{1-x^2}\right) \, dx=\int_{-1}^1
   \exp (a x) \sum _{n=0}^{\infty } \frac{\left(b \sqrt{1-x^2}\right)^n}{n!} \, dx=\sum _{n=0}^{\infty } \int_{-1}^1 \frac{\exp (a
   x) \left(b \sqrt{1-x^2}\right)^n}{n!} \, dx=\sum _{n=0}^{\infty } \frac{2^{\frac{1}{2}-\frac{n}{2}} a^{-\frac{1}{2}-\frac{n}{2}}
   b^n \pi  I_{\frac{1+n}{2}}(a)}{\Gamma \left(\frac{1}{2}+\frac{n}{2}\right)}=\sqrt{\frac{2}{a}} \pi  \sum _{n=0}^{\infty }
   \frac{(2 a)^{-\frac{n}{2}} b^n I_{\frac{1+n}{2}}(a)}{\Gamma \left(\frac{1}{2}+\frac{n}{2}\right)}$$
For integral to solve I'm use a CAS.
