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The ratio of branching fractions and the difference in $CP$ asymmetries ofthe decays $B^+\to J/\psi \pi^+$ and $B^+\to J/\psi K^+$ are measured using adata sample of $pp$ collisions collected by the LHCb experiment, correspondingto an integrated luminosity of $3~{{\,\mbox{fb}^{-1}}}$ at centre-of-massenergies of 7 and $8~\mathrm{\,Te\kern -0.1em V}$. The results are\begin{equation*} {\mathcal{B}(B^+\to J/\psi \pi^+)\over\mathcal{B}(B^+\toJ/\psi K^+)}=(3.83\pm0.03\pm0.03)\times10^{-2}\,, \end{equation*}\begin{equation*} \mathcal{A}^{CP}(B^+\to J/\psi \pi^+)-\mathcal{A}^{CP}(B^+\toJ/\psi K^+)=(1.82\pm0.86\pm0.14)\times10^{-2}, \end{equation*} where the firstuncertainties are statistical and the second are systematic. Combining thisresult with a recent LHCb measurement of $\mathcal{A}^{CP}(B^+\to J/\psi K^+)$provides the most precise estimate to date of $CP$ violation in the decay$B^+\to J/\psi \pi^+$, \begin{equation*} \mathcal{A}^{CP}(B^+\to J/\psi \pi^+)=(1.91\pm0.89\pm0.16)\times10^{-2}. \end{equation*}

Measurement of the ratio of branching fractions and difference in $CP$ asymmetries of the decays $B^+\to J/\psi\pi^+$ and $B^+\to J/\psi K^+$