# Integrating the exponential of a periodic function

I am wondering whether the following integral

$$Y = \int_{x_0}^{x_1} \exp{\left(a\cos x+b\sin x\right)}\,dx,$$

has a known closed form for given $a,b,x_0$ and $x_1$.

Since the exponential of the periodic function $a\cos x+b\sin x$ is also periodic, I know that this integral can be computed under a period, and the solution in terms of a modified Bessel function can be found here.

But, what about for general integral limits?

Thanks for any hint!

• – Count Iblis Jul 21 '18 at 16:39
• Be more specific, please. – A. Pongrácz Jul 21 '18 at 18:06
• @A.Pongrácz, I did not get your point. – Alex Silva Jul 21 '18 at 18:25
• @CountIblis, Thanks for the paper. :) – Alex Silva Jul 22 '18 at 18:35