A measurement of the W boson pair production cross section in proton-proton collisions at s√=8 TeV is presented. The data collected with the CMS detector at the LHC correspond to an integrated luminosity of 19.4fb-1. The W+W− candidates are selected from events with two charged leptons, electrons or muons, and large missing transverse energy. The measured W+W− cross section is 60.1±0.9(stat)±3.2(exp)±3.1(theo)±1.6(lumi)\,pb=60.1±4.8\,pb, consistent with the standard model prediction. The W+W− cross sections are also measured in two different fiducial phase space regions. The normalized differential cross section is measured as a function of kinematic variables of the final-state charged leptons and compared with several perturbative QCD predictions. Limits on anomalous gauge couplings associated with dimension-six operators are also given in the framework of an effective field theory. The corresponding 95 % confidence level intervals are −5.7<cWWW/Λ2<5.9TeV−2, −11.4<cW/Λ2<5.4TeV−2, −29.2<cB/Λ2<23.9TeV−2, in the HISZ basis.