$\int \sqrt{\frac{1+x}{1-x}} \, dx = $

If $\int \frac{dx}{\sin^3 x + \cos^3 x} = A \log \left|\frac{\sqrt{2}+t}{\sqrt{2}-t}\right| + B \tan^{-1}(t) + c$,then $\left(\frac{B}{A}, t\right) =$

If $\int(1+x) \log \left(1+x^2\right) d x=\left(x+\frac{x^2}{2}+\frac{1}{2}\right) \log \left(1+x^2\right)+g(x)+C$,then $g(x)=$

$\int \frac{\sec ^2 x}{\sin ^7 x} d x - \int \frac{7}{\sin ^7 x} d x =$

Let $\int {{\sec }^{ - 1}}\left[ { - {\sin }^2x} \right]dx = f(x) + C$,(valid for $x \neq 0$) where $[k]$ denotes the greatest integer less than or equal to $k$ and $f(0) = 0$. Then the value of ${\left( {f\left( {\frac{8}{{\pi x}}} \right)} \right)''}$ at $x = 2$ is (where $'$ denotes the derivative).

Find the integral of the function $\frac{1}{\cos (x-a) \cos (x-b)}$.