  The origin of high-temperature superconductivity in cuprates is still an
unresolved issue. Among the most likely candidates for mediating the Cooper
pair condensate are spin fluctuations and the electron-phonon interaction.
While the former have long been proposed to be responsible for various
observables in the superconducting state, the latter has recently been shown to
produce unconventional gap symmetries when vertex corrections are
self-consistently taken into account. Here, we develop multichannel Eliashberg
theory to incorporate both pairing mechanisms. Solving self-consistently the
full-bandwidth, anisotropic Eliashberg equations for a cuprate model system we
find the characteristic $d_{x^2 -y^2}$ symmetry of the superconducting gap and
a reasonable order of magnitude for $T_c$ ($50-120$ K). We further find that
both mechanisms support an unconventional $d$-wave symmetry of the order
parameter, yet the electron-phonon interaction is chiefly responsible for the
Cooper pairing and high $T_c$, whereas the spin fluctuations have a suppressing
effect on the gap magnitude and critical temperature.
