Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis.
In smaller systems, it's possible to directly insert this equation, but as systems grow, this approach becomes unmanageable. Nevertheless, it can be shown that the density operator can be represented in a diagonal format if it's based on an overcomplete basis.