Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for a few special cases. In this work, we present and compare different ways to construct high order symplectic schemes for general Hamiltonian systems that can be split in three integrable parts. We use these techniques to numerically solve the equations of motion for a simple toy model, as well as the disordered discrete nonlinear Schr\"odinger equation. We thereby compare the efficiency of symplectic and non-symplectic integration methods. Our results show that the new symplectic schemes are superior to the other tested methods, with respect to both long term energy conservation and computational time requirements.