If $\int \frac{x}{\sqrt{x+1}+\sqrt{x-1}} d x=A(x)(x+1)^{\frac{3}{2}}+B(x)(x-1)^{\frac{3}{2}}+C$,then $A(x)+B(x)=$

Evaluate the integral $\int \frac{\log x - (\log x)^2 + x^2}{x^3} dx$ (where $C$ is the constant of integration).

Observe the following statements :
$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$
$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$
Then which of the following is true?

Let $f(x) = \int \frac{dx}{(3+4x^2) \sqrt{4-3x^2}}$,$|x| < \frac{2}{\sqrt{3}}$. If $f(0) = 0$ and $f(1) = \frac{1}{\alpha \beta} \tan^{-1}\left(\frac{\alpha}{\beta}\right)$,where $\alpha, \beta > 0$,then $\alpha^2 + \beta^2$ is equal to $.........$.

If $\int \operatorname{cosec}^5 x \, dx = \alpha \cot x \operatorname{cosec} x \left(\operatorname{cosec}^2 x + \frac{3}{2}\right) + \beta \log_e \left|\tan \frac{x}{2}\right| + C$,where $\alpha, \beta \in R$ and $C$ is the constant of integration,then the value of $8(\alpha + \beta)$ is equal to:

If $\int \frac{5 \tan x}{\tan x-2} \, dx = x + a \log |\sin x - 2 \cos x| + c$ (where $c$ is a constant of integration),then the value of $a$ is