  In this work, the anisotropic quantum Rabi model with different coupling
strengths of the rotating-wave and counter-rotating wave terms is studied by
using two kinds of extended coherent states (ECS). By the first kind of ECS, we
can derive a so-called $G$-function, by which both the regular and exceptional
solutions can be given. The exceptional solution are just corresponding to the
crossing points of two energy levels with different parities, so is doubly
degenerate. By the second kind of ECS, a general scheme for the eigensolutions
is derived analytically in a unified way. The zero-order approximation is just
the adiabatic approximation, and the first-order approximation is actually a
generalized rotating-wave approximation. The algebraic formulae for the
eigensolutions are given explicitly in two approximations. The generalized
rotating-wave approximations work well in a wide range of two different
coupling strengths and the qubit detunings.
