  The Hamiltonian of a linearly driven two-level system, or qubit, in the
standard rotating frame contains non-commuting terms that oscillate at twice
the drive frequency, $\omega$, rendering the task of analytically finding the
qubit's time evolution nontrivial. The application of the rotating wave
approximation (RWA), which is suitable only for drives whose amplitude, or
envelope, $H_1(t)$, is small compared to $\omega$ and varies slowly on the time
scale of $1/\omega$, yields a simple Hamiltonian that can be integrated
relatively easily. We present a series of corrections to the RWA Hamiltonian in
$1/\omega$, resulting in an effective Hamiltonian whose time evolution is
accurate also for time-dependent drive envelopes in the regime of strong
driving, i.e., for $|H_1(t)| \lesssim \omega$. By extending the Magnus
expansion with the use of a Taylor series we introduce a method that we call
the Magnus-Taylor expansion, which we use to derive a recurrence relation for
computing the effective Hamiltonian. We then employ the same method to derive
kick operators, which complete our theory for non-smooth drives. The time
evolution generated by our kick operators and effective Hamiltonian, both of
which depend explicitly on the envelope and its time derivatives, agrees with
the exact time evolution at periodic points in time. For the leading
Hamiltonian correction we obtain a term proportional to the first derivative of
the envelope, which competes with the Bloch-Siegert shift.
