Modern flexible production systems benefit from collaborative robot systems that support to teach robot configurations by hand to quickly implement collision-free motions. However, appropriate interpolation schemes that allow a fast and smooth motion through this sequence of waypoints is often not part of robot control systems. Instead, only point-to-point motions or a set of predefined curves like circular motions are supported. We propose an approach to improve the efficiency of finding a time parameterization with cubic splines that computes minimum trajectory durations while respecting actuator limits and $C^2$-continuity. Compared to the existing method, our contribution consists in the analytical gradients of the underlying nonlinear optimization problem and thus we are able to propose an efficient solution approach using state-of-the-art optimization solvers. We validate our implementation in simulation and in experiments, and benchmark it with the three trajectory generation methods implemented in MoveIt!.