$$\int_0^{2\pi} \int_0^{2\pi} \min \{ x-y, 2\pi- (x-y) \} e^{a\cos(x)} e^{a\cos(y)} dx dy, \qquad a\in\mathbb R$$
I tried to find the integral like this integral (https://math.stackexchange.com/q/2346355)
but I  couldn't, I look forward to your experience.