<s>[INST] Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $ How to evaluate the integral
$$\int_{25\pi / 4}^{53\pi / 4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx\ ?$$
 [/INST] For any integrable function $f:\mathbb{R}\to\mathbb{R}$ with period $T$ and any $a,b\in\mathbb{R}$ we have
$$
\int_{b}^{b+T} f(x)dx=\int_{a}^{a+T}f(x)dx\tag{1}
$$
And as the consequence for $k\in\mathbb{N}$
$$
k\int_{b}^{b+T} f(x)dx=\int_{a}^{a+kT}f(x)dx\tag{2}
$$
Also for any integrable $f$ we have
$$
\int\limits_{-c}^c f(x)dx=\int\limits_{0}^{c}(f(x)+f(-x))dx\tag{3}
$$
Now we turn to evaluation
$$
\begin{align}
I_1
&=\int_{25\pi/4}^{29\pi/4}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx\\
&=\int_{\pi/4+3\cdot 2\pi}^{5\pi/4+3\cdot 2\pi}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx\\
&\overset{(1)}{=}\int_{\pi/4}^{5\pi/4}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx\\
&=\int\limits_{\pi/4}^{3\pi/4}\left(\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}+\frac{1}{(1+2^{\cos (x+\pi/2)})(1+2^{\sin (x+\pi/2)})}\right)dx\\
&=\int\limits_{\pi/4}^{3\pi/4}\frac{dx}{1+2^{\cos x}}\\
&=\int\limits_{\pi/4}^{\pi/2}\left(\frac{1}{1+2^{\cos x}}+\frac{1}{1+2^{\cos (\pi-x)}}\right)dx\\
&=\int\limits_{\pi/4}^{\pi/2}dx\\
&=\frac{\pi}{4}\\
\end{align}
$$
$$
\begin{align}
I_2
&=\int_{29\pi/4}^{53\pi/4}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx\\
&=\int_{29\pi/4}^{29\pi/4+3\cdot 2\pi}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx\\
&\overset{(2)}{=}3\int_{-\pi}^{\pi}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx\\
&\overset{(3)}{=}3\int\limits_{0}^{\pi} \left(\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}+\frac{1}{(1+2^{\cos (-x)})(1+2^{\sin (-x)})}\right)dx\\
&=3\int\limits_{0}^{\pi}\frac{dx}{1+2^{\cos x}}\\
&=3\int\limits_{-\pi/2}^{\pi/2}\frac{dx}{1+2^{\cos (x+\pi/2)}}\\
&\overset{(3)}{=}3\int\limits_{0}^{\pi/2}\left(\frac{1}{1+2^{\cos (x+\pi/2)}}+\frac{1}{1+2^{\cos (-x+\pi/2)}}\right)dx\\
&=3\int\limits_{0}^{\pi/2}=\frac{3\pi}{2}\\
\end{align}
$$
Finally,
$$
I=I_1+I_2=\frac{7\pi}{4}
$$
</s>