We discuss analytic-numeric methods for solving stiff scenarios in computer animation. Classical explicit numerical integration schemes have the shortcoming that step sizes are limited by the highest frequency that occurs within the solution spectrum of the governing equations, while implicit methods suffer from an inevitable and mostly uncontrollable artificial viscosity that often leads to non-physical behavior. To overcome these specific detriments, an appropriate class of so-called exponential integrators that solves the stiff part of the governing equations by employing a closed-form solution is presented. With these techniques, up to three orders of magnitude greater time steps can be handled compared to conventional methods, and, at the same time, a tremendous increase in overall long-term stability is achieved. This advantageous behavior is demonstrated across a broad spectrum of stiff scenarios that include deformable solids, trusses, and textiles, including damping, collision responses, and friction. To realize an efficient and physically accurate simulation of stiff fiber-based systems such as human hair, wool infills, and brushes, an appropriate approach for the physically accurate simulation of densely packed fiber assemblies is presented.