In this paper we compare two numerical methods to integrate Riemannian cubic polynomials on the Stiefel manifold \(\textbf{St}_{n,k}\) . The first one is the adjusted de Casteljau algorithm, and the second one is a symplectic integrator constructed through discretization maps. In particular, we choose the cases of \(n=3\) together with \(k=1\) and \(k=2\) . The first case is diffeomorphic to the sphere and the quasi-geodesics appearing in the adjusted de Casteljau algorithm are actually geodesics. The second case is an example where we have a pure quasi-geodesic different from a geodesic. We provide a numerical comparison of both methods and discuss the obtained results to highlight the benefits of each method.