Indefinite Integration

Let \(\alpha \in\left(0, \frac{\pi}{2}\right)\) be fixed. If the integral
\(\int \frac{\tan x+\tan \alpha}{\tan x-\tan \alpha} \mathrm{d} x=\mathrm{A}(x) \cos 2 \alpha+\mathrm{B}(x) \sin 2 \alpha+\mathrm{c},\)
(where \(\mathrm{c}\) is a constant of integration), then functions \(\mathrm{A}(x)\) and \(\mathrm{B}(x)\) are respectively