  We construct an integral $p$-adic cohomology that compares with rigid
cohomology after inverting $p$. Our approach is based on the log-Witt
differentials of Hyodo-Kato and log-\'etale motives of
Binda-Park-{\O}stv{\ae}r. In case $k$ satisfies resolutions of singularities,
we moreover prove that it agrees with the "good" integral $p$-adic cohomology
of Ertl-Shiho-Sprang: from this we deduce some interesting motivic properties
and a K\"unneth formula for the $p$-adic cohomology of Ertl-Shiho-Sprang.
