$ \int_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx $ any hints? I am struggling with following integral.
$$
\int_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx
$$
A small hint is enough.

Here is my approach by using the well-known property : $\arctan(f(x)) + \arctan\left(\frac{1}{f(x)}\right) = \frac{\pi}{2}$. One has:
$$\int_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx = \int_{- \frac{\pi}{4}}^{0} \tan^{-1}(e^{\tan(x)}) dx+ \int_{0}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx$$
$$ \overset{t = - x} =\int_{0}^{\frac{\pi}{4}} \tan^{-1}(e^{-\tan(t)}) dt + \int_{0}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx$$ $$
$$ $$=\int_{0}^{\frac{\pi}{4}} \tan^{-1}(e^{-\tan(x)})+ \tan^{-1}(e^{\tan(x)}) dx= \frac{\pi}{2}\int_{0}^{\frac{\pi}{4}} dx = \frac{\pi^2}{8} $$
