  We present a systematic method for evaluation of perturbation observables in
non-canonical single-field inflation models within the slow-roll approximation,
which allied with field redefinitions enables predictions to be established for
a wide range of models. We use this to investigate various non-canonical
inflation models, including Tachyon inflation and DBI inflation. The Lambert
$W$ function will be used extensively in our method for the evaluation of
observables. In the Tachyon case, in the slow-roll approximation the model can
be approximated by a canonical field with a redefined potential, which yields
predictions in better agreement with observations than the canonical
equivalents. For DBI inflation models we consider contributions from both the
scalar potential and the warp geometry. In the case of a quartic potential, we
find a formula for the observables under both non-relativistic and relativistic
behaviour of the scalar DBI inflaton. For a quadratic potential we find two
branches in the non-relativistic case, determined by the competition of model
parameters, while for the relativistic case we find consistency with results
already in the literature. We present a comparison to the latest Planck
satellite observations. Most of the non-canonical models we investigate,
including the Tachyon, are better fits to data than canonical models with the
same potential, but we find that DBI models in the slow-roll regime have
difficulty in matching the data.
