TITLE: Find $\int_{-\pi/2}^{\pi/2} \frac{\cos^3 x}{e^x+1} dx$
QUESTION [6 upvotes]: Find the value of the following integral
$$
\int_{-\pi/2}^{\pi/2} \frac{\cos^3 x}{e^x+1} dx
$$
I tried to use the twin integral with sin and get some result but nothing obvious came out. Also, Mathematica gives a very bizarre result! Any ideas how to solve analytically?

REPLY [3 votes]: Like Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$.,
if $\displaystyle I=\int_{-a}^a\dfrac{g(x)\ dx}{b^x+1}$
$\displaystyle I=\int_{-a}^a\dfrac{g(a-a-x)\ dx}{b^{a-a-x}+1}$
If $g(x)=g(-x)$ i.e., an even function,
$\displaystyle I=\int_{-a}^ag(x)\ dx$
Here $g(x)=\cos^3x=\dfrac{\cos3x+3\cos x}4$