Integrate $$\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.} $$

If for a real number $$y$$, $$\left[ y \right]$$ is the greatest integer less than or
equal to $$y$$, then the value of the integral $$\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx} $$ is