Observation of $D^+toetaetapi^+$ and improved measurement of $D^{0(+)}toetapi^+pi^{-(0)}$

Using an $e^+e^-$ annihilation data sample corresponding to an integrated luminosity of $2.93,rm fb^{-1}$ collected at the center-of-mass energy of 3.773,GeV with the BESIII detector, we measure the absolute branching fractions of $D^+toetaetapi^+$, $D^+toetapi^+pi^0$, and $D^0toetapi^+pi^-$ to be $(2.96 pm 0.24 pm 0.13)times 10^{-3}$, $(2.23 pm 0.15 pm 0.11)times 10^{-3}$, and $(1.20 pm 0.07 pm 0.04)times 10^{-3}$, respectively, where the first uncertainties are statistical and the second ones systematic. The $D^+toetaetapi^+$ decay is observed for the first time and the branching fractions of $D^{+(0)}toetapi^+pi^{0(-)}$ are measured with much improved precision. In addition we test for $CP$ asymmetries in the separated charge-conjugate branching fractions; no evidence of $CP$ violation is found.