Measurement of the branching fractions of $D_{s}^{+}rightarrow etaX$ and $D_{s}^{+}rightarrow etarho^{+}$ in $e^+e^-to D^+_{s}D^-_{s}$

We study $D_{s}^{+}$ decays to final states involving the $eta$ with a 482$,$pb$^{-1}$ data sample collected at $sqrt{s}$ = 4.009$,$GeV with the mbox{BESIII} detector at the BEPCII collider. We measure the branching fractions $mathcal{B}(D^+_{s}rightarrow etaX)$ = (8.8$pm$1.8$pm$0.5)$%$ and $mathcal{B}(D_{s}^{+}rightarrow etarho^{+})$ = ($5.8pm1.4pm0.4$)$%$ where the first uncertainty is statistical and the second is systematic. In addition, we estimate an upper limit on the non-resonant branching ratio $mathcal{B}(D_{s}^{+}rightarrow etapi^+pi^0)<5.1%$ at the 90$%$ confidence level. Our results are consistent with CLEOs recent measurements and help to resolve the disagreement between the theoretical prediction and CLEOs previous measurement of $mathcal{B}(D_{s}^{+}rightarrow etarho^{+})$.