  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 $4.178$ GeV 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 $CP$ asymmetry is observed in the
decay rates $D_s^\pm\to\mu^\pm\nu_\mu$ and $D_s^\pm\to\tau^\pm\nu_\tau$:
$A_{CP}(\mu^\pm\nu) = (-1.2\pm2.7)\%$ and $A_{CP}(\tau^\pm\nu) =
(+2.9\pm4.9)\%$. Constraining our measurement to the Standard Model expectation
of lepton universality ($R=9.75$), we find the more precise results $\cal
B(D_s^+\to \tau^+\nu_\tau) = (5.22\pm0.10\pm 0.14)\times10^{-2}$ and
$A_{CP}(\tau^\pm\nu_\tau) = (-0.1\pm2.1)\%$. Combining our results with inputs
external to our analysis, we determine the $c\to s$ quark mixing matrix
element, $D_s^+$ decay constant, and ratio of the decay constants: $|V_{cs}| =
0.973\pm0.009\pm0.014$, $f_{D^+_s} = 249.9\pm2.4\pm3.5~\text{MeV}$, and
$f_{D^+_s}/f_{D^+} = 1.232\pm0.035$.
