Inverse cosec summation

Let a series be defined as $\csc ^{ -1 }{ \sqrt { 10 } } +\quad \csc ^{ -1 }{ \sqrt { 50 } + } \csc ^{ -1 }{ \sqrt { 170 } }$ , .+. $\csc ^{ -1 }{ \sqrt { \left( { n }^{ 2 }+1 \right) \left( { n }^{ 2 }+2n+2 \right) } }$ ...... infinity = $\frac { \alpha \pi }{ \beta }$ then find the value of $\alpha +\beta$