The processes $X(3872)to D^{*0}bar{D^{0}}+c.c.,~gamma J/psi,~gamma psi(2S),$ and $gamma D^{+}D^{-}$ are searched for in a $9.0~rm fb^{-1}$ data sample collected at center-of-mass energies between $4.178$ and $4.278$ GeV with the BESIII detector. We observe $X(3872)to D^{*0}bar{D^{0}}+c.c.$ and find evidence for $X(3872)togamma J/psi$ with statistical significances of $7.4sigma$ and $3.5sigma$, respectively. No evident signals for $X(3872)togammapsi(2S)$ and $gamma D^{+}D^{-}$ are found, and upper limit on the relative branching ratio $R_{gamma psi} equivfrac{mathcal{B}(X(3872)togammapsi(2S))}{mathcal{B}(X(3872)togamma J/psi)}<0.59$ is set at 90$%$ confidence level. Measurements of branching ratios relative to decay $X(3872)topi^+pi^- J/psi$ are also reported for decays $X(3872)to D^{*0}bar{D^{0}}+c.c., ~gammapsi(2S),~gamma J/psi$, $gamma D^{+}D^{-}$, as well as the non-$D^{*0}bar{D}^{0}$ three-body decays $pi^0 D^{0}bar{D}^{0}$ and $gamma D^{0}bar{D}^{0}$.
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