If $\int \frac{2 \cos x+3 \sin x}{4 \cos x+5 \sin x} dx = \left(\frac{23}{41}\right) x + K \log |4 \cos x+5 \sin x| + c$,then $K$ is equal to

$\int \sec x \tan^3 x \, dx = $

If $\int \frac{dx}{1-\sin^4 x} = A \tan x + B \tan^{-1}(\sqrt{2} \tan x) + C$,then $A^2 - B^2 =$

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

If $\int \frac{1}{x^4+8 x^2+9} d x = \frac{1}{k} \left[ \frac{1}{\sqrt{14}} \tan^{-1}(f(x)) - \frac{1}{\sqrt{2}} \tan^{-1}(g(x)) \right] + c$, then $\sqrt{\frac{k}{2} + f(\sqrt{3}) + g(1)} =$

$\int \frac{3\cos x + 3\sin x}{4\sin x + 5\cos x} \, dx = $