A measurement is reported of the ratio of branching fractions R(J/ψ)=B(B_{c}^{+}→J/ψτ^{+}ν_{τ})/B(B_{c}^{+}→J/ψμ^{+}ν_{μ}), where the τ^{+} lepton is identified in the decay mode τ^{+}→μ^{+}ν_{μ}ν[over ¯]_{τ}. This analysis uses a sample of proton-proton collision data corresponding to 3.0 fb^{-1} of integrated luminosity recorded with the LHCb experiment at center-of-mass energies of 7 and 8 TeV. A signal is found for the decay B_{c}^{+}→J/ψτ^{+}ν_{τ} at a significance of 3 standard deviations corrected for systematic uncertainty, and the ratio of the branching fractions is measured to be R(J/ψ)=0.71±0.17(stat)±0.18(syst). This result lies within 2 standard deviations above the range of central values currently predicted by the standard model.