Hybrid integration and acceleration techniques are used to speed up the electron guiding centre orbit calculations in a toroidal axisymmetric magnetized plasma in the presence of a dc electric field. Acceleration of the computation is introduced to bridge the gap between the two different time scales: the rapid circulations of the electron around the magnetic axis and the relatively infrequent collisions and slow response to the electric field. The constants of motion method is utilized in correcting the particle position after each guiding centre step, which makes long time steps in the integration possible. The method described is applied to the simulation of reverse runaway electrons in a tokamak plasma. It is shown that in some cases fairly large average acceleration factors (100–10000) are acceptable: the statistics for an ensemble of electrons remain correct and the transition of a single electron through the trapped orbit velocity cone is properly described. A good agreement is found with 2-D Fokker-Planck calculations in the straight cylinder approximation for a cold plasma. The finite temperature and toroidal effects on the reverse runaway rate are calculated, and the adverse effects of runaway electrons on current ramp-up in tokamaks are discussed.