If $\int \frac{1}{\cos 4x \cos 2x} dx = \frac{1}{2\sqrt{2}} \log \left(\frac{1+f(x)}{1-f(x)}\right) - \frac{1}{2} \log g(x) + C$,then find the value of $g\left(\frac{\pi}{6}\right) - \sqrt{2} f\left(\frac{\pi}{6}\right)$.

The value of $\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} \,d x$ is equal to (where $C$ is an arbitrary constant.)

$\int \frac{3\cos x + 3\sin x}{4\sin x + 5\cos x} \, dx = $

If $I_n = \int \frac{t^{2n}}{1+t^2} dt$,then $I_{n+1} =$

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$,if $\int \left( \left( \frac{x}{e} \right)^{2x} + \left( \frac{e}{x} \right)^{2x} \right) \log_{e} x \, dx = \frac{1}{\alpha} \left( \frac{x}{e} \right)^{\beta x} - \frac{1}{\gamma} \left( \frac{e}{x} \right)^{\delta x} + C$,where $e = \sum_{n=0}^{\infty} \frac{1}{n!}$ and $C$ is the constant of integration,then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

If $\int f(x) \sin x \cos x \, dx = \frac{1}{2(b^2 - a^2)} \log f(x) + c$,where $c$ is the constant of integration,then $f(x)$ is