\int{-\frac{t^2+1}{2t}\cdot{\frac{t^2+1}{2t^2}dt}}=-\frac{1}{4}\int{\frac{t^4+2t^2+1}{t^3}}dt=-\frac{1}{4}\int{(t+\frac{2}{t}+t^{-3})}dt=
-\frac{t^2}{8}-\frac{1}{2}ln|t|-\frac{1}{4}\cdot{\frac{-1}{2t^2}}+C=-\frac{t^2}{8}-\frac{1}{2}ln|t|+\frac{1}{8t^2}+C