  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 $H$.
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.
