If $\int \frac{\mathrm{d} x}{x^4+5 x^2+4}=\mathrm{A} \tan ^{-1} x+\mathrm{B} \tan ^{-1} \frac{x}{2}+\mathrm{c}$ where c is a constant of integration, then

$$ \int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x} d x= $$

If $\mathrm{f}(x)=\frac{\sin ^2 x}{1+\cot x}+\frac{\cos ^2 x}{1+\tan x}$, then the value of $\mathrm{f}^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

If $y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leqslant x<\frac{\pi}{2}$, then $y^{\prime}\left(\frac{\pi}{6}\right)=$