$\int_{0}^{\pi/2} \text{arctanh}(\sin x) \text{arctan}(a \tan(x)) \cos(x) \ dx$

Question

Are you kind to let me know the way? By the way, don't you have a "curiosity" tag? $$\int_{0}^{\pi/2} \text{arctanh}(\sin x) \text{arctan}(a \tan(x)) \cos(x) \ dx, \quad a>0$$

Answer 1

$$\forall a>0,\quad\int_{0}^{\pi/2}\text{artanh}(\sin x)\,\arctan(a\,\tan(x))\cos(x)\ dx=\\\frac\pi2\Re\left(\ln\left(2\,a^{-1}+2\right)-\frac{\ln\left(2\sqrt{a^{-4}-a^{-2}}+2\,a^{-2}-1\right)}{2 \sqrt{1-a^2}}\right),$$ where $\Re$ denotes the real part.