  We present a semiclassical prediction of regular-to-chaotic tunneling in
systems with a mixed phase space, including the effect of a nonlinear resonance
chain. We identify complex paths for direct and resonance-assisted tunneling in
the phase space of an integrable approximation with one nonlinear resonance
chain. We evaluate the resonance-assisted contribution analytically and give a
prediction based on just a few properties of the classical phase space. For the
standard map excellent agreement with numerically determined tunneling rates is
observed. The results should similarly apply to ionization rates and quality
factors.
