  We introduce the cylindrical module $A \natural \mathcal{H}$, where
$\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over
$\mathcal{H}$. We show that there exists an isomorphism between
$\mathsf{C}_{\bullet}(A^{op} \rtimes \mathcal{H}^{cop})$ the cyclic module of
the crossed product algebra $A^{op} \rtimes \mathcal{H}^{cop} $, and $\Delta(A
\natural \mathcal{H}) $, the cyclic module related to the diagonal of $A
\natural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it
follows that $\mathsf{C}_{\bullet}(A \rtimes \mathcal{H}) \simeq \Delta(A^{op}
\natural \mathcal{H}^{cop})$. When $S$ is invertible, we approximate
$HC_{\bullet}(A \rtimes \mathcal{H})$ by a spectral sequence and give an
interpretation of $ \mathsf{E}^0, \mathsf{E}^1$ and $\mathsf{E}^2 $ terms of
this spectral sequence.
