$\int \frac{(\tan^{-1} x)^3}{1 + x^2} \, dx = $

If $\int \frac{e^x}{\sqrt{e^{2x}+4e^x+13}} dx = \log \left|e^x+2+\sqrt{e^{2x}+4e^x+13}\right|+c$,(where $c$ is the constant of integration),then the value of $a$ in the expression $\log \left|e^{ax}+2+\sqrt{e^{2x}+4e^x+13}\right|+c$ is equal to

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

If $\int \frac{\cos x}{\sqrt{4 \sin ^2 x+4 \sin x+5}} d x=\frac{1}{2} \sinh ^{-1}(f(x))+C$, then find $2 f(x)$.

$\int 2x \cos^3(x^2) \sin(x^2) \, dx = $

$\int \frac{1}{\sqrt{2x-x^2}} dx = $