If $\int \frac{1 - (\cot x)^{2021}}{\tan x + (\cot x)^{2022}} dx = \frac{1}{A} \log |(\sin x)^{2023} + (\cos x)^{2023}| + c$,then $A = . . . . . .$

$\int \frac{3\cos x + 2\sin x}{4\sin x + 5\cos x} dx = A \{23x + 2\ln |4\sin x + 5\cos x|\} + c$,then $A$ and $f(x)$ are

$\int \frac{dx}{\sin(x-a) \cos(x-b)} = $

$\int \frac{dx}{\sin^6 x + \cos^6 x} = $

$\int_0^{\pi /2} {\left( {\frac{\theta }{{\sin \theta }}} \right)^2 d\theta = } $

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