$\int \frac{dx}{\tan x+\cot x+\sec x+\operatorname{cosec} x} = $

$\int \frac{dx}{4+3 \cot x} = $

$\int \frac{x^5+x}{x^8+1} dx =$

$\int \frac{x^4-1}{x^2 \sqrt{x^4+x^2+1}} \, dx =$

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 $\int \sqrt{\frac{x - 5}{x - 7}} dx = A \sqrt{x^2 - 12 x + 35} + \log |x - 6 + \sqrt{x^2 - 12 x + 35}| + C$,then $A = . . . . . .$