This article is devoted to the comparison of numerical integration methods for nonsmooth multibody dynamics with joints, unilateral contacts and impacts in an industrial context. With an event–driven strategy, the smooth dynamics, which is integrated between two events, may be equivalently formulated as a Differential Algebraic Equation (DAE) of index 1, 2 or 3. It is well-known that these reformulations are no longer equivalent when a numerical time–integration technique is used. The drift-off effect and the stability of the numerical scheme strongly depend on the index of the formulation. But, besides the standard properties of accuracy and stability of the DAE solvers, the event–driven context imposes some further requirements that are crucial for a robust and efficient event-driven strategy. In this article, several state–of–the–art numerical time integration methods for each formulation are compared: the generalized-α scheme for index-3 formulation and stabilized index-2 formulation, (Partitioned) Runge–Kutta Half–Explicit Method of order 5 (HEM5 and PHEM56) for index-2 DAEs with projection techniques, and Runge-Kutta explicit scheme of order 5, the Dormand-Prince scheme (DOPRI5), for index-1 DAEs with projection techniques (MDOP5). We compare these schemes in terms of efficiency, violation of the constraints and the way they handle stiff dynamics on numerous industrial benchmarks, where a CAD software is in this loop. One of the major conclusions is that the index-2 DAEs solvers prove to be better than other schemes to maintain low violations at position and acceleration levels. The best compromise allows us to design efficient event–driven solvers. When the dynamics is stiff, implicit schemes outperform explicit and half-explicit methods whichare sometimes unable to compute the dynamics when the system’s frequency range is wide. Furthermore, in industrial context, some solvers fail to reproduce the properties that they enjoy in theory. This is particularly true for half-explicit schemes when the Jacobian of the constraints has not full rank.