We report measurements of the flux-integrated $bar{ u}_mu$ and $bar{ u}_mu+ u_mu$ charged-current cross-sections on water and hydrocarbon targets using the T2K anti-neutrino beam, with a mean neutrino energy of 0.86 GeV. The signal is defined as the (anti-)neutrino charged-current interaction with one induced $mu^pm$ and no detected charged pion nor proton. These measurements are performed using a new WAGASCI module recently added to the T2K setup in combination with the INGRID Proton module. The phase space of muons is restricted to the high-detection efficiency region, $p_{mu}>400~{rm MeV}/c$ and $theta_{mu}<30^{circ}$, in the laboratory frame. Absence of pions and protons in the detectable phase space of $p_{pi}>200~{rm MeV}/c$ and $theta_{pi}<70^{circ}$, and $p_{rm p}>600~{rm MeV}/c$ and $theta_{rm p}<70^{circ}$ is required. In this paper, both of the $bar{ u}_mu$ cross-sections and $bar{ u}_mu+ u_mu$ cross-sections on water and hydrocarbon targets, and their ratios are provided by using DAgostini unfolding method. The results of the integrated $bar{ u}_mu$ cross-section measurements over this phase space are $sigma_{rm H_{2}O},=,(1.082pm0.068(rm stat.)^{+0.145}_{-0.128}(rm syst.)) times 10^{-39}~{rm cm^{2}/nucleon}$, $sigma_{rm CH},=,(1.096pm0.054(rm stat.)^{+0.132}_{-0.117}(rm syst.)) times 10^{-39}~{rm cm^{2}/nucleon}$, and $sigma_{rm H_{2}O}/sigma_{rm CH} = 0.987pm0.078(rm stat.)^{+0.093}_{-0.090}(rm syst.)$. The $bar{ u}_mu+ u_mu$ cross-section is $sigma_{rm H_{2}O} = (1.155pm0.064(rm stat.)^{+0.148}_{-0.129}(rm syst.)) times 10^{-39}~{rm cm^{2}/nucleon}$, $sigma_{rm CH},=,(1.159pm0.049(rm stat.)^{+0.129}_{-0.115}(rm syst.)) times 10^{-39}~{rm cm^{2}/nucleon}$, and $sigma_{rm H_{2}O}/sigma_{rm CH},=,0.996pm0.069(rm stat.)^{+0.083}_{-0.078}(rm syst.)$.
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