  In this paper we develop a family of three 8-step methods, optimized for the
numerical integration of oscillatory ordinary differential equations. We have
nullified the phase-lag of the methods and the first r derivatives, where
r=1,2,3. We show that with this new technique, the method gains efficiency with
each derivative of the phase-lag nullified. This is the case for the
integration of both the Schrodinger equation and the N-body problem. A local
truncation error analysis is performed, which, for the case of the Schrodinger
equation, also shows the connection of the error and the energy, revealing the
importance of the zero phase-lag derivatives. Also the stability analysis shows
that the methods with more derivatives vanished, have a bigger interval of
periodicity.
