If $\int \frac{(x^2-1)}{(x+1)^2 \sqrt{x(x^2+x+1)}} dx = A \tan^{-1}\left(\sqrt{\frac{x^2+x+1}{x}}\right) + C$, where $C$ is a constant, then $A$ equals to

Evaluate the integral $\int \frac{\log x - (\log x)^2 + x^2}{x^3} dx$ (where $C$ is the constant of integration).

$\begin{aligned} & \int \frac{x \, dx}{\sqrt[15]{\left(1+x^2\right)^{12}\left(2+x^2\right)^{18}}}=\alpha\left(\frac{1+x^2}{2+x^2}\right)^{1 / n}+C \Rightarrow \\ & \frac{n}{\alpha}= \end{aligned}$

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).

$\int \frac{x - \sin x}{1 - \cos x} dx = $

If $\int \frac{3 \cos x-2 \sin x}{4 \sin x+5 \cos x} d x=A \log |5 \cos x+4 \sin x|+B x+c$,then $A$ and $B$ are