We report new measurements of the branching fraction $cal B(D_s^+to ell^+ u)$, where $ell^+$ is either $mu^+$ or $tau^+(topi^+bar{ u}_tau)$, based on $6.32$ fb$^{-1}$ of electron-positron annihilation data collected by the BESIII experiment at six center-of-mass energy points between $4.178$ GeV and $4.226$ GeV. Simultaneously floating the $D_s^+tomu^+ u_mu$ and $D_s^+totau^+ u_tau$ components yields $cal B(D_s^+to tau^+ u_tau) = (5.21pm0.25pm0.17)times10^{-2}$, $cal B(D_s^+to mu^+ u_mu) = (5.35pm0.13pm0.16)times10^{-3}$, and the ratio of decay widths $R=frac{Gamma(D_s^+to tau^+ u_tau)}{Gamma(D_s^+to mu^+ u_mu)} = 9.73^{+0.61}_{-0.58}pm 0.36$, where the first uncertainties are statistical and the second systematic. No evidence of $CP$ asymmetry is observed in the decay rates $D_s^pmtomu^pm u_mu$ and $D_s^pmtotau^pm u_tau$: $A_{CP}(mu^pm u) = (-1.2pm2.7)%$ and $A_{CP}(tau^pm u) = (+2.9pm4.9)%$. Constraining our measurement to the Standard Model expectation of lepton universality ($R=9.75$), we find the more precise results $cal B(D_s^+to tau^+ u_tau) = (5.22pm0.10pm 0.14)times10^{-2}$ and $A_{CP}(tau^pm u_tau) = (-0.1pm2.1)%$. Combining our results with inputs external to our analysis, we determine the $cto s$ quark mixing matrix element, $D_s^+$ decay constant, and ratio of the decay constants: $|V_{cs}| = 0.973pm0.009pm0.014$, $f_{D^+_s} = 249.9pm2.4pm3.5~text{MeV}$, and $f_{D^+_s}/f_{D^+} = 1.232pm0.035$.
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