Solving Tricky Integral: How to Proceed Further?

[QwertyPoiuyt]

I was trying the evaluate the integral

$$\int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})} dx$$ from

I have since manipulated this integral into $$\int_{\pi/4}^{5\pi/4}\frac{7}{(1+2^{\sin x})(1+2^{\cos x})} dx$$
Any help on how to proceed further would be appreciated.

The value of the original integral is I believe $$7\pi/4$$The Ineresting thing is I had solved this in the past but don't have a clue on how to go further now..

 

[gmax137]

The value of the original integral is I believe

7π/4​

I did a quick numerical integration on your first integral, I get ~3.57. So I don't think that's correct.

 

[QwertyPoiuyt]

I did a quick numerical integration on your first integral, I get ~3.57. So I don't think that's correct.

You can check on wolfram alpha It returns the value 5.49774 which is the value I quoted