  We place observational constraints on slow-variation single-field
inflationary models by carrying out the cosmological Monte Carlo simulation
with the recent data of Planck combined with the WMAP large-angle polarization,
baryon acoustic oscillations, and ACT/SPT temperature data. Our analysis covers
a wide variety of models with second-order equations of motion-- including
potential-driven slow-roll inflation, non-minimally coupled models, running
kinetic couplings, Brans-Dicke theories, potential-driven Galileon inflation,
field-derivative couplings to the Einstein tensor, and k-inflation. In the
presence of running kinetic exponential couplings, covariant Galileon terms,
and field-derivative couplings, the tensor-to-scalar ratio of the self-coupling
potential gets smaller relative to that in standard slow-roll inflation, but
the models lie outside the 68 % CL observational contour. We also show that
k-inflation models can be tightly constrained by adding the bounds from the
scalar non-Gaussianities. The small-field inflationary models with asymptotic
flat Einstein-frame potentials in the regime phi >> M_{pl} generally fit the
data very well. These include the models such as Kahler-moduli inflation,
non-minimally coupled Higgs inflation, and inflation in Brans-Dicke theories in
the presence of the potential V(phi)=3M^2 (phi-M_{pl})^2/4 with the Brans-Dicke
parameter omega_{BD}<O(1) (which covers the Starobinsky's model
f(R)=R+R^2/(6M^2) as a special case).
