The first measurement that relates semileptonic and hadronic decay rates of the $B_c^+$ meson is performed using protonproton collision data corresponding to 1.0 fb$^{1}$ of integrated luminosity collected with the LHCb detector. The measured value of the ratio of branching fractions, ${\cal B}(B_c^+ \to J/\psi \pi^+)/{\cal B}(B_c^+\to J/\psi\mu^+\nu_\mu)=0.0469 \pm 0.0028 (stat.) \pm 0.0046 (syst.)$, is at the lower end of available theoretical predictions.
Invariantmass distribution of $B_c^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \pi^+$ candidates (black data points). The maximum likelihood fit of the $B_c^+$ signal is superimposed (blue solid line). Individual fit components are also shown: (dashed blue line) the signal, (red longdashed line) the background and (green dotted line) $B_c^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K^+$ feeddown. 
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Distribution of $m_{ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \mu}$ for $B_c^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \mu^+\nu_\mu$ candidates selected in simulated event samples of (blue filled points) the signal, (green filled points) the $B_c^+$ feeddown and (red filled squares) the $B_{u,d,s}$ backgrounds. Relative normalization is derived from the fit to the data described later in the text. The part of the spectrum included in the fit is indicated with a vertical dashed black line. The $B_c^+$ feeddown distribution is also shown after magnifying its normalization by a factor of ten (green dashed histogram). 
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Invariantmass distribution of $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \mu^+$ pairs from $B_c^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \mu^+\nu_\mu$ candidates (black data points) for (top left) the data, (bottom left) $B_c^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \mu^+\nu_\mu$ signal simulation, (top right) $B_{u,d,s}\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} X$ background simulation and (bottom right) $B_c^+$ feeddown simulation. The unbinned maximum likelihood fit of the $B_c^+$ signal is superimposed (blue solid line). Individual fit components are also shown: (blue shortdashed line) the signal, (red longdashed line) the background and (green dotted line) $B_c^+$ feeddown. 
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The measured value of $\cal R$ (horizontal solid line) and its $\pm1\sigma$ uncertainty band (dashed lines) compared to the predictions (diamonds). A nonrelativistic reduction of the BetheSalpeter equation is used in the predictions of Chang {\slshape et al.\/} [17], ElHady \eta [20], and Colangelo \eta [21], while the latter also utilizes heavy quark symmetry. A lightfront constituent quark model is used by Anisimov \eta [18] and Ke \eta [25]. QCD sum rules are used by Kiselev \eta [22], a relativistic quasipotential Schr\"odinger model is used by Ebert \eta [23], and a relativistic constituent quark model in used by Ivanov \eta [24]. 
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Animated gif made out of all figures. 
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Values of the parameters affecting the estimate of the feeddown fraction in the fit to the $m_{ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \mu}$ distribution. For $B_c^+\rightarrow \chi_{cJ}\mu^+\nu_\mu$, $\sum_{J=0,1,2} R_{{\it f J}} \cal B_{\rm casc {\it f J}}$ is listed. 
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Summary of systematic uncertainties. The total systematic errors are obtained by adding in quadrature the individual contributions. 
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Created on 10 July 2020.