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

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 ....... .

$\int \frac{2 \cos 2 x}{(1+\sin 2 x)(1+\cos 2 x)} d x=$

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

$\int \frac{2 x+2}{\sqrt{x^2-4 x-5}} d x$ is equal to

If $\int {\frac{{\csc^2 x}}{{{{\left( {\csc x + \cot x} \right)}^{\frac{9}{2}}}}}\,dx} = {\left( {\csc x - \cot x} \right)^{\frac{7}{2}}}\left( {\frac{1}{\alpha } + \frac{{{{\left( {\csc x - \cot x} \right)}^2}}}{{11}}} \right) + C$ (where $C$ is the constant of integration and $\alpha \in N$),then $\alpha$ is: