Measurements of the branching fractions for the semi-leptonic decays $D_s^{+}\to \varphi e^{+}{\nu }_e$, $\varphi {\mu }^{+}{\nu }_{\mu }$, $\eta {\mu }^{+}{\nu }_{\mu }$ and ${\eta }^{\mathrm{\prime }}{\mu }^{+}{\nu }_{\mu }$

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 $D_{s}^{+}\to \phi e^{+}\nu_{e}$,$\phi \mu^{+}\nu_{\mu}$,$\eta \mu^{+}\nu_{\mu}$and$\eta’\mu^{+}\nu_{\mu}$tobe${\mathcal B}(D_{s}^{+}\to\phi e^{+}\nu_{e})=(2.26\pm0.45\pm0.09)$\mathrm{\%},${\mathcal B}(D_{s}^{+}\to\phi \mu^{+}\nu_{\mu})=(1.94\pm0.53\pm0.09)$\mathrm{\%},${\mathcal B}(D_{s}^{+}\to\eta \mu^{+}\nu_{\mu})=(2.42\pm0.46\pm0.11)$\mathrm{\%}and${\mathcal B}(D_{s}^{+}\to\eta’\mu^{+} \nu_{\mu}) = (1.06\pm0.54\pm0.07)$%, where the first and second uncertainties are statistical and systematic, respectively. The branching fractions for the three semi-muonic decays $D_s^+\to\phi \mu^+\nu_\mu, \eta \mu^+\nu_\mu$and$\eta’ \mu^+\nu_\mu$ are determined for the first time and that of $D_s^{+}\to \varphi e^{+}{\nu }_e$ is consistent with the world average value within uncertainties.