In this paper, we study the effects of symmetrization by the implicit midpoint rule (IMR) and the implicit trapezoidal rule
(ITR) on the numerical solution of ordinary differential equations. We extend the study of the well-known formula of Gragg
to a two-step symmetrizer and compare the efficiency of their use with the IMR and ITR. We present the experimental results
on nonlinear problem using variable stepsize setting and the results show greater efficiency of the two-step symmetrizers
over the one-step symmetrizers of IMR and ITR.