  In this paper we present two optimized eight-step symmetric implicit methods
with phase-lag order ten and infinite (phase-fitted). The methods are
constructed to solve numerically the radial time-independent Schr\"odinger
equation with the use of the Woods-Saxon potential. They can also be used to
integrate related IVPs with oscillating solutions such as orbital problems. We
compare the two new methods to some recently constructed optimized methods from
the literature. We measure the efficiency of the methods and conclude that the
new method with infinite order of phase-lag is the most efficient of all the
compared methods and for all the problems solved.
