  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 $D^+\to \eta \mu^+\nu_\mu$ 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 ${\mathcal B}_{D^+\to \eta e^+\nu_e}$, the
ratio of the two branching fractions is determined to be ${\mathcal B}_{D^+\to
\eta \mu^+\nu_\mu}/{\mathcal B}_{D^+\to \eta e^+\nu_e}=0.91\pm0.13$, 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 $c\to d$ Cabibbo-Kobayashi-Maskawa matrix
element $|V_{cd}|$ to be $f_{+}^\eta (0)|V_{cd}|=0.087\pm0.008_{\rm
stat}\pm0.002_{\rm syst}$. Taking the input of $|V_{cd}|$ from the global fit
in the standard model, we determine $f_{+}^\eta (0)=0.39\pm0.04_{\rm
stat}\pm0.01_{\rm syst}$. On the other hand, using the value of $f_+^{\eta}(0)$
calculated in theory, we find $|V_{cd}|=0.242\pm0.022_{\rm stat}\pm0.006_{\rm
syst}\pm0.033_{\rm theory}$.
