  Shortcuts to adiabaticity (STA) is a general methodology for speeding up
adiabatic quantum protocols, and has many potential applications in quantum
information processing. Unfortunately, analytically constructing STAs for
systems having complex interactions and more than a few levels is a challenging
task. This is usually overcome by assuming an idealized Hamiltonian (e.g., only
a limited subset of energy levels are retained, and the rotating-wave
approximation (RWA) is made). Here, we develop an $analytic$ approach that
allows one to go beyond these limitations. Our method is general and results in
analytically-derived pulse shapes that correct both non-adiabatic errors as
well as non-RWA errors. We also show that our approach can yield pulses
requiring a smaller driving power than conventional non-adiabatic protocols. We
show in detail how our ideas can be used to analytically design high-fidelity
single-qubit "tripod" gates in a realistic superconducting fluxonium qubit.
