  The decays $B_c^+ \to J/\psi D_s^+$ and $B_c^+ \to J/\psi D_s^{*+}$ are
studied with the ATLAS detector at the LHC using a dataset corresponding to
integrated luminosities of 4.9 fb$^{-1}$ and 20.6 fb$^{-1}$ of $pp$ collisions
collected at centre-of-mass energies $\sqrt{s} = 7$ TeV and 8 TeV,
respectively. Signal candidates are identified through $J/\psi\to\mu^+\mu^-$
and $D_s^{(*)+}\to\phi\pi^+(\gamma/\pi^0)$ decays. With a two-dimensional
likelihood fit involving the $B_c^+$ reconstructed invariant mass and an angle
between the $\mu^+$ and $D_s^+$ candidate momenta in the muon pair rest frame,
the yields of $B_c^+ \to J/\psi D_s^+$ and $B_c^+ \to J/\psi D_s^{*+}$, and the
transverse polarisation fraction in $B_c^+ \to J/\psi D_s^{*+}$ decay are
measured. The transverse polarisation fraction is determined to be
$\Gamma_{\pm\pm}(B_c^+ \to J/\psi D_s^{*+})/\Gamma(B_c^+ \to J/\psi D_s^{*+}) =
0.38 \pm 0.23 \pm 0.07$, and the derived ratio of the branching fractions of
the two modes is $\mathcal{B}_{B_c^+ \to J/\psi D_s^{*+}}/\mathcal{B}_{B_c^+
\to J/\psi D_s^+} = 2.8 \,^{+1.2}_{-0.8} \pm 0.3$, where the first error is
statistical and the second is systematic. Finally, a sample of $B_c^+\to
J/\psi\pi^+$ decays is used to derive the ratios of branching fractions
$\mathcal{B}_{B_c^+ \to J/\psi D_s^+}/\mathcal{B}_{B_c^+ \to J/\psi \pi^+} =
3.8 \pm 1.1 \pm 0.4 \pm 0.2$ and $\mathcal{B}_{B_c^+ \to J/\psi
D_s^{*+}}/\mathcal{B}_{B_c^+ \to J/\psi \pi^+} = 10.4 \pm 3.1 \pm 1.5 \pm 0.6$,
where the third error corresponds to the uncertainty of the branching fraction
of $D_s^+\to\phi(K^+K^-)\pi^+$ decay. The available theoretical predictions are
generally consistent with the measurement.
