Measurements of the branching fractions for the semi-leptonic decays $D^+_stophi e^{+} u_{e}$, $phi mu^{+} u_{mu}$, $eta mu^{+} u_{mu}$ and $etamu^{+} u_{mu}$

By analyzing 482 pb$^{-1}$ of $e^+e^-$ collision data collected at the center-of-mass energy $sqrt s=4.009$ GeV with the BESIII detector, we measure the %absolute branching fractions for the semi-leptonic decays $D_{s}^{+}to phi e^{+} u_{e}$, $phi mu^{+} u_{mu}$, $eta mu^{+} u_{mu}$ and $etamu^{+} u_{mu}$ to be ${mathcal B}(D_{s}^{+}tophi e^{+} u_{e})=(2.26pm0.45pm0.09)$%, ${mathcal B}(D_{s}^{+}tophi mu^{+} u_{mu})=(1.94pm0.53pm0.09)$%, ${mathcal B}(D_{s}^{+}toeta mu^{+} u_{mu})=(2.42pm0.46pm0.11)$% and ${mathcal B}(D_{s}^{+}toetamu^{+} u_{mu}) = (1.06pm0.54pm0.07)$%, where the first and second uncertainties are statistical and systematic, respectively. The branching fractions for the three semi-muonic decays $D_s^+tophi mu^+ u_mu, eta mu^+ u_mu$ and $eta mu^+ u_mu$ are determined for the first time and that of $D^+_sto phi e^+ u_e$ is consistent with the world average value within uncertainties.