$$\int\limits_0^{2\pi} \int\limits_0^{2\pi} {\min \{ x-y, 2\pi- (x-y) \} e^{a\cos(x)} e^{a\cos(y)}} dx dy, \qquad a\in\mathbb R. $$ I tried to find the value of the integral following the method proposed in this example: however I didn't succeed so I posted the question here looking forward to your experience.