Current dynamical models of Cosserat rods often use the finite element method limited by computational efficiency or the finite difference method in a Cartesian framework with a compromise to accuracy. We employ the finite difference method in a geometric framework to develop solutions that are both computationally efficient and accurate. A numerical study is conducted on various backward-differentiation discretization and Runge-Kutta-Munthe-Kaas integration schemes, focusing on their accuracy and computational efficiency. Case studies are conducted on a single-degree-of-freedom joint actuated Cosserat rod to mitigate additional sources of undesired error from the numerical analysis, e.g. multi-body interactions, moving base dynamics, etc. The proposed geometric integrators are demonstrated to improve solution accuracy compared to the published finite difference models. The presented solution is parameterization-free and also computationally efficient with the potential for use in real-time applications, e.g., model-based control of soft manipulators.