By analyzing a data sample corresponding to an integrated luminosity of $2.93~mathrm{fb}^{-1}$ collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, we measure for the first time the absolute branching fraction of the $D^+to eta mu^+ u_mu$ decay to be ${mathcal B}_{D^+to eta mu^+ u_mu}=(10.4pm1.0_{rm stat}pm0.5_{rm syst})times 10^{-4}$. Using the world averaged value of ${mathcal B}_{D^+to eta e^+ u_e}$, the ratio of the two branching fractions is determined to be ${mathcal B}_{D^+to eta mu^+ u_mu}/{mathcal B}_{D^+to eta e^+ u_e}=0.91pm0.13$, which agrees with the theoretical expectation of lepton flavor universality within uncertainty. Here, the uncertainty is the sum in quadrature of the statistical and systematic uncertainties. By studying the differential decay rates in five four-momentum transfer intervals, we obtain the product of the hadronic form factor $f^{eta}_{+}(0)$ and the $cto d$ Cabibbo-Kobayashi-Maskawa matrix element $|V_{cd}|$ to be $f_{+}^eta (0)|V_{cd}|=0.087pm0.008_{rm stat}pm0.002_{rm syst}$. Taking the input of $|V_{cd}|$ from the global fit in the standard model, we determine $f_{+}^eta (0)=0.39pm0.04_{rm stat}pm0.01_{rm syst}$. On the other hand, using the value of $f_+^{eta}(0)$ calculated in theory, we find $|V_{cd}|=0.242pm0.022_{rm stat}pm0.006_{rm syst}pm0.033_{rm theory}$.
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