\int \frac{1+u^2}{u^3} \, \mathrm{d}u = \int \Bigg{(}\frac{1}{u^3}+\frac{1}{u} \Bigg{)} \, \mathrm{d}u = \int \frac{\mathrm{d}u}{u^3} + \int \frac{\mathrm{d}u}{u}=C+\log|u| - \frac{1}{2u^2}