  We explore thick branes in $f(R,\phi,X)$ gravity. We obtain the linear tensor
perturbation equation of $f(R,\phi,X)$ branes and show that the branes are
stable against the tensor perturbations under the condition of $\frac{\partial
f(R,\phi,X)}{\partial R}>0$. In order to obtain thick brane solutions of the
fourth-order field equations in this theory, we employ the reconstruction
technique. We get exact solutions of the specific $f(R,\phi,X)$ thick brane
generated by a non-canonical scalar field. It is shown that the zero mode of
the graviton for the thick brane is localized under certain conditions. This
implies that the four-dimensional Newtonian potential is recovered on the
brane. The effects of the Kaluza-Klein modes of the graviton for the
$f(R,\phi,X)$ thick brane are also discussed.
