The simulation of multi-body systems with frictional contacts is a fundamental tool for many fields, such as robotics, computer graphics, and mechanics. Hard frictional contacts are particularly troublesome to simulate because they make the differential equations stiff, calling for computationally demanding implicit integration schemes. We suggest to tackle this issue by using exponential integrators, a long-standing class of integration schemes (first introduced in the 60’s) that in recent years has enjoyed a resurgence of interest. We show that this scheme can be easily applied to multi-body systems subject to stiff viscoelastic contacts, producing accurate results at lower computational cost than classic explicit schemes. In our tests with quadruped and biped robots, our method demonstrated stable behaviors with large time steps (10 ms) and stiff contacts ($10^5$ N/m). Its excellent properties, especially for fast and coarse simulations, make it a valuable candidate for many applications in robotics, such as simulation, Model Predictive Control, Reinforcement Learning, and controller design.