TITLE: Trigonometric integral: $\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$
QUESTION [8 upvotes]: Is it possible to evaluate the following in a closed form?
$$\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$

I found the above definite integral at I&S but the solution is not given. I have tried various methods but none of them lead me to the solution, I am honestly out of ideas for this one. 
Any help is appreciated. Thanks!

REPLY [3 votes]: $$I_{a,b}=\int_{a\pi}^{b\pi} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$
$$I_{25/4,53/4}=I_{0,7}=3I_{0,2}+I_{0,1}$$