$\int \frac{e^{x}(1+x)}{\cos ^{2}(x e^{x})} d x$ equals

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 $2 f(x)$.

$\int \frac{\log \sqrt{x}}{3 x} d x$ is equal to

Integrate the function: $\frac{1}{x \sqrt{ax - x^{2}}} \quad \left[ \text{Hint: } x = \frac{a}{t} \right]$

The integral $\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} d x$ is equal to (Where $C$ is a constant of integration).

Let $n \ge 2$ be a natural number and $0 < \theta < \frac{\pi}{2}$. Then $\int \frac{(\sin^n \theta - \sin \theta)^{\frac{1}{n}} \cos \theta}{\sin^{n+1} \theta} d\theta$ is equal to