Inthispaper,analmostP-stabletwo-stepsixth-orderHybridmethodwithphase-lagoforderinfinityandaclassexpliciteighth-orderObreckoffmethodswithphase-lagoforder10-24aredevelopedforthenumericalintegrationofthespecialsecond-orderperiodicinitial-valueproblems.Thesemethodshavetheadvantageofhigheralgebraicorderandconsiderablysmallerphase-tagcomparedwithsomemethodsin[1-6].Numericalexamplesindicatethatthesenewmethodsaremoreaccuratethanmethodsdevelopedby[1-6].