$\int \frac{dx}{\sin(x-a) \cos(x-b)} = $

For real numbers $\alpha, \beta, \gamma$ and $\delta,$ if $\int \frac{\left(x^{2}-1\right)+\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)} d x =\alpha \log _{e}\left(\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)\right) +\beta \tan ^{-1}\left(\frac{\gamma\left(x^{2}-1\right)}{x}\right)+\delta \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)+C$ where $C$ is an arbitrary constant,then the value of $10(\alpha+\beta \gamma+\delta)$ is equal to ....... .

Let a function $h(x)$ be defined as $h(x) = 0$, for all $x \ne 0$. Also $\int_{-\infty}^{\infty} h(x) \cdot f(x) \, dx = f(0)$, for every function $f(x)$. Then the value of the definite integral $\int_{-\infty}^{\infty} h'(x) \cdot \sin x \, dx$ is

$\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x=$

If $\int \frac{1}{(x-2)^5(x-1)^4} d x=\sum_{r=-4}^{-1} A_r\left(\frac{x-2}{x-1}\right)^r+\sum_{r=1}^3 A_r\left(\frac{x-2}{x-1}\right)^r+B f(x)$, then $f(x)=$

$\int \sqrt{x^{2}-8 x+7} \, dx$ is equal to