I need to calculate the following integral: $$I = \int_0^{\pi} \sin(x) e^{-\sqrt{1-\cos(x)}-\sqrt{1+\sin(x)}} dx$$ The numerical value is 0.22110767725647466402... So far, I tried Maple and Inverse Symbolic Calculator (which found nothing). The plot of the integrand
Any hints about showing that it doesn't have a closed form solution will also be very much appreciated.