  We obtain a mixed complex simpler than the canonical one the computes the
type cyclic homologies of a crossed product with invertible cocycle
$A\times_{\rho}^f H$, of a weak module algebra $A$ by a weak Hopf algebra $H$
whose unit cocommutes. This complex is provided with a filtration. The spectral
sequence of this filtration generalizes the spectral sequence obtained in
\cite{CGG}. When $f$ takes its values in a separable subalgebra of $A$ that
satisfies suitable conditions, the above mentioned mixed complex is provided
with another filtration, whose spectral sequence generalize the Feigin-Tsygan
spectral sequence.
