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A measurement of the differential branching fraction of the decay${B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}}$ is presented together witha determination of the S-wave fraction of the $K^+\pi^-$ system in the decay$B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$. The analysis is based on$pp$-collision data corresponding to an integrated luminosity of 3\,fb$^{-1}$collected with the LHCb experiment. The measurements are made in bins of theinvariant mass squared of the dimuon system, $q^2$. Precise theoreticalpredictions for the differential branching fraction of $B^{0}\rightarrowK^{\ast}(892)^{0}\mu^{+}\mu^{-}$ decays are available for the $q^2$ region$1.1<q^2<6.0\,{\rm GeV}^2/c^4$. In this $q^2$ region, for the $K^+\pi^-$invariant mass range $796 < m_{K\pi} < 996\,{\rm MeV}/c^2$, the S-wave fractionof the $K^+\pi^-$ system in $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$decays is found to be \begin{equation*}  F_{\rm S} = 0.101\pm0.017({\rm stat})\pm0.009 ({\rm syst}), \end{equation*}and the differential branching fraction of $B^{0}\rightarrowK^{\ast}(892)^{0}\mu^{+}\mu^{-}$ decays is determined to be \begin{equation*}  {\rm d}\mathcal{B}/{\rm d} q^2 = (0.342_{\,-0.017}^{\,+0.017}({\rmstat})\pm{0.009}({\rm syst})\pm0.023({\rm norm}))\times 10^{-7}c^{4}/{\rmGeV}^{2}. \end{equation*} The differential branching fraction measurementspresented are the most precise to date and are found to be in agreement withStandard Model predictions.

Measurements of the S-wave fraction in $B^{0}\rightarrow  K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays and the $B^{0}\rightarrow  K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ differential branching fraction