\int t\frac{2tdt}{(t^2+1)^2}=2\int \frac{t^2dt}{(t^2+1)^2}=2\int \frac{t^2+1-1}{(t^2+1)^2}dt=2\biggl(\int \frac{t^2+1}{(t^2+1)^2}dt - \int \frac{1}{(t^2+1)^2}dt\biggr) = 2\biggl(\int \frac{1}{t^2+1}dt -  \int \frac{1}{(t^2+1)^2}dt\biggr)