The azimuthal anisotropic flow of identified and unidentified charged particles has been systematically studied in Cu+Au collisions at $sqrt{s_{_{NN}}}$ = 200 GeV for harmonics $n=$ 1-4 in the pseudorapidity range $|eta|<1$. The directed flow in Cu+Au collisions is compared with the rapidity-odd and, for the first time, the rapidity-even components of charged particle directed flow in Au+Au collisions at $sqrt{s_{_{NN}}}$ = 200~GeV. The slope of the directed flow pseudorapidity dependence in Cu+Au collisions is found to be similar to that in Au+Au collisions, with the intercept shifted toward positive $eta$ values, i.e., the Cu-going direction. The mean transverse momentum projected onto the spectator plane, $langle p_xrangle$, in Cu+Au collision also exhibits approximately linear dependence on $eta$ with the intercept at about $etaapprox-0.4$, closer to the rapidity of the Cu+Au system center-of-mass. The observed dependencies find natural explanation in a picture of the directed flow originating partly due the tilted source and partly due to the rapidity dependent asymmetry in the initial density distribution. Charge-dependence of the $langle p_xrangle$ was also observed in Cu+Au collisions, indicating an effect of the initial electric field created by charge difference of the spectator protons in two colliding nuclei. The rapidity-even component of directed flow in Au+Au collisions is close to that in Pb+Pb collisions at $sqrt{s_{_{NN}}}$ = 2.76 TeV, indicating a similar magnitude of dipole-like fluctuations in the initial-state density distribution. Higher harmonic flow in Cu+Au collisions exhibits similar trends to those observed in Au+Au and Pb+Pb collisions and is qualitatively reproduced by a viscous hydrodynamic model and a multi-phase transport model. For all harmonics with $nge2$ we observe an approximate scaling of $v_n$ with the number of constituent quarks.
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