Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evalu- ations per step which is the most economical in computational cost that can be found in the literature. The methods have high ampli cation and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our e ort.