$\int \frac{1 + \tan x}{x + \log \sec x} \, dx = $

If $\int {\frac{{({x^2} - 1)\,dx}}{{({x^4} + 3{x^2} + 1)\,{{\tan }^{ - 1}}\left( {\frac{{{x^2} + 1}}{x}} \right)}}} = \ln | f(x) | + C$,then $f(x)$ is:

Integrate the function $\frac{\sin x}{(1+\cos x)^{2}}$.

$\int \frac{dx}{\cos^2(x) + \sin(2x)} = $

If $f(x) = \sqrt{\tan x}$ and $g(x) = \sin x \cdot \cos x$,then $\int \frac{f(x)}{g(x)} dx$ is equal to (where $C$ is a constant of integration).

For $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,if $y(x) = \int \frac{\operatorname{cosec} x + \sin x}{\operatorname{cosec} x \sec x + \tan x \sin^2 x} \, dx$ and $\lim_{x \rightarrow (\frac{\pi}{2})^-} y(x) = 0$,then $y\left(\frac{\pi}{4}\right)$ is equal to: