$\int \frac{1}{1 + \sin^2 x} \, dx = $

If $\int \frac{1-(\cot x)^{2019}}{\tan x+(\cot x)^{2020}} dx = \frac{1}{n} \ln |(f(x))^n + (g(x))^n| + c$, then the value of $n[(f(x))^4 + (g(x))^4]_{x=\frac{\pi}{3}}$ is:

$\int \frac{x^2+1}{x^4+7 x^2+1} d x$ is equal to

If $\int \frac{1}{x} \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=2 f(x)-2 \operatorname{Sin}^{-1} \sqrt{x}+c$, then $f(x)=$

If $\int \frac{3 e^x-7 e^{-x}}{7 e^x+3 e^{-x}} d x=K x+L \log \left(e^{-2 x}+\frac{7}{3}\right)+C$,then $K+L=$

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).