The N-body problem has become one of the hottest topics in the fields of computational dynamics and cosmology. The large dynamical range in some astrophysical problems led to the use of adaptive time steps to integrate particle trajectories, however, the search of optimal strategies is still challenging. We quantify the performance of the hierarchical time step integrator Hamiltonian Splitting (HamSp) for collisionless multistep simulations. We compare with the constant step Leap-Frog (LeapF) integrator and the adaptive one (AKDK). Additionally, we explore the impact of different time step assigning functions. There is a computational overhead in HamSp however there are two interesting advantages: choosing a convenient time-step function may compensate and even turn around the efficiency compared with AKDK. We test both reversibility and time symmetry. The symmetrized nature of the HamSp integration is able to provide time-reversible integration for medium time scales and overall deliver better energy conservation for long integration times, and the linear and angular momentum are preserved at machine precision. We address the impact of using different integrators in astrophysical systems. We found that in most situations both AKDK and HamSp are able to correctly simulate the problems. We conclude that HamSp is an attractive and competitive alternative to AKDK, with, in some cases, faster and with better energy and momentum conservation. The use of recently discussed Bridge splitting techniques with HamSp may allow to reach considerably high efficiency.
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