Solving Tricky Integral: How to Proceed Further?

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SUMMARY

The integral $$\int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})} dx$$ has been manipulated into the form $$\int_{\pi/4}^{5\pi/4}\frac{7}{(1+2^{\sin x})(1+2^{\cos x})} dx$$. The original integral's value is confirmed to be $$\frac{7\pi}{4}$$, although numerical integration suggests a different result of approximately 3.57. Wolfram Alpha provides a more accurate value of 5.49774, indicating a discrepancy in the initial calculations.

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QwertyPoiuyt
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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..
 
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QwertyPoiuyt said:
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.
 
gmax137 said:
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
 

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