I think I thought of this integral a little over a year ago and I just haven't been able to do it. I really want to and everytime I sit down with it I learn a little bit more (or atleast I think I do) and then I get really stumped and give up.
The integral in question is
$$I=\int_{0}^{\pi}\frac{e^{-\sin(x)}}{e^{-\sin(x)}+e\sin(x)}dx$$
Which can be rewritten into
$$I=2\int_{0}^{\pi/2}\frac{e^{-\sin(x)}}{e^{-\sin(x)}+e\sin(x)}dx$$
by making note of the fact that the function enclosed in the integral is even.
One other thing to note is that if we consider the $esin(x)$ term in the denominator and write a "new" integral with this term replacing the $e^{-\sin(x)}$ term in the numerator we get the following
$$J=\int_{0}^{\pi}\frac{e\sin(x)}{e^{-\sin(x)}+e\sin(x)}dx.$$
Taking the sum of $I$ and $J$ we, of course, get the answer $\pi$, meaning that a solution to either one would give a solution to the other (hopefully this helps in some way). I have a decent background and level of understanding with many of the more popular "advanced" integration techniques and special functions, so don't be afraid to write a really scary looking solution. Finally, I genuinely don't think there is a nice closed-form solution for this integral, but I don't know how to show that and am really hoping there is a closed-form.
Best of luck with your solutions.