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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 withthe BESIII detector, we measure for the first time the absolute branchingfraction of the $D^+\to \eta \mu^+\nu_\mu$ decay to be ${\mathcal B}_{D^+\to\eta \mu^+\nu_\mu}=(10.4\pm1.0_{\rm stat}\pm0.5_{\rm syst})\times 10^{-4}$.Using the world averaged value of ${\mathcal B}_{D^+\to \eta e^+\nu_e}$, theratio of the two branching fractions is determined to be ${\mathcal B}_{D^+\to\eta \mu^+\nu_\mu}/{\mathcal B}_{D^+\to \eta e^+\nu_e}=0.91\pm0.13$, whichagrees with the theoretical expectation of lepton flavor universality withinuncertainty. Here, the uncertainty is the sum in quadrature of the statisticaland systematic uncertainties. By studying the differential decay rates in fivefour-momentum transfer intervals, we obtain the product of the hadronic formfactor $f^{\eta}_{+}(0)$ and the $c\to d$ Cabibbo-Kobayashi-Maskawa matrixelement $|V_{cd}|$ to be $f_{+}^\eta (0)|V_{cd}|=0.087\pm0.008_{\rmstat}\pm0.002_{\rm syst}$. Taking the input of $|V_{cd}|$ from the global fitin the standard model, we determine $f_{+}^\eta (0)=0.39\pm0.04_{\rmstat}\pm0.01_{\rm syst}$. On the other hand, using the value of $f_+^{\eta}(0)$calculated in theory, we find $|V_{cd}|=0.242\pm0.022_{\rm stat}\pm0.006_{\rmsyst}\pm0.033_{\rm theory}$.

First Observation of $D^+\to \eta\mu^+\nu_\mu$ and Measurement of its Decay Dynamics