A calculus problem by Saarthak Marathe

Calculus Level 5

$\large \displaystyle I= \int^{1}_{0} \dfrac { \tan^{-1}(\sqrt{2+x^2})}{(1+x^2)\sqrt{2+x^2}} \,dx$

Find the value of $\lfloor{1000I}\rfloor$ .

Notation: $\lfloor{\cdot}\rfloor$ denotes the floor function (greatest integer function).