  The $CP$ asymmetry in the mixing of $B_s^0$ and $\overline{B}{}_s^0$ mesons
is measured in proton-proton collision data corresponding to an integrated
luminosity of 3.0 fb$^{-1}$, recorded by the LHCb experiment at centre-of-mass
energies of 7 and 8 TeV. Semileptonic $B_s^0$ and $\overline{B}{}_s^0$ decays
are studied in the inclusive mode $D_s^{\mp} \mu^{\pm} \nu_{\mu} X$ with the
$D_s^{\mp}$ mesons reconstructed in the $K^+ K^- \pi^{\mp}$ final state.
Correcting the observed charge asymmetry for detection and background effects,
the $CP$ asymmetry is found to be $a_{\text{sl}}^s = (0.39 \pm 0.26 \pm 0.20)
\% $, where the first uncertainty is statistical and the second systematic.
This is the most precise measurement of $a_{\text{sl}}^s$ to date. It is
consistent with the prediction from the Standard Model and will constrain new
models of particle physics.
