  We report new measurements of the branching fraction $\cal B(D_s^+\to
\ell^+\nu)$, where $\ell^+$ is either $\mu^+$ or
$\tau^+(\to\pi^+\bar{\nu}_\tau)$, based on $6.32$ fb$^{-1}$ of
electron-positron annihilation data collected by the BESIII experiment at six
center-of-mass energy points between $4.178$ and $4.226$ GeV. Simultaneously
floating the $D_s^+\to\mu^+\nu_\mu$ and $D_s^+\to\tau^+\nu_\tau$ components
yields $\cal B(D_s^+\to \tau^+\nu_\tau) = (5.21\pm0.25\pm0.17)\times10^{-2}$,
$\cal B(D_s^+\to \mu^+\nu_\mu) = (5.35\pm0.13\pm0.16)\times10^{-3}$, and the
ratio of decay widths $R=\frac{\Gamma(D_s^+\to \tau^+\nu_\tau)}{\Gamma(D_s^+\to
\mu^+\nu_\mu)} = 9.73^{+0.61}_{-0.58}\pm 0.36$, where the first uncertainties
are statistical and the second systematic. No evidence of ${\it CP}$ asymmetry
is observed in the decay rates $D_s^\pm\to\mu^\pm\nu_\mu$ and
$D_s^\pm\to\tau^\pm\nu_\tau$: $A_{\it CP}(\mu^\pm\nu) = (-1.2\pm2.5\pm1.0)\%$
and $A_{\it CP}(\tau^\pm\nu) = (+2.9\pm4.8\pm1.0)\%$. Constraining our
measurement to the Standard Model expectation of lepton universality
($R=9.75$), we find the more precise results $\cal B(D_s^+\to \tau^+\nu_\tau) =
(5.22\pm0.10\pm 0.14)\times10^{-2}$ and $A_{\it CP}(\tau^\pm\nu_\tau) =
(-0.1\pm1.9\pm1.0)\%$. Combining our results with inputs external to our
analysis, we determine the $c\to \bar{s}$ quark mixing matrix element, $D_s^+$
decay constant, and ratio of the decay constants to be $|V_{cs}| =
0.973\pm0.009\pm0.014$, $f_{D^+_s} = 249.9\pm2.4\pm3.5~\text{MeV}$, and
$f_{D^+_s}/f_{D^+} = 1.232\pm0.035$, respectively.
