  Let k be a field, A a unitary associative k-algebra and V a k-vector space
endowed with a distinguished element 1_V. We obtain a mixed complex, simpler
that the canonical one, that gives the Hochschild, cyclic, negative and
periodic homology of a crossed product E:=A#_f V, in the sense of Brzezinski.
We actually work in the more general context of relative cyclic homology.
Specifically, we consider a subalgebra K of A that satisfies suitable
hypothesis and we find a mixed complex computing the Hochschild, cyclic,
negative and periodic homology of E relative to K. Then, when E is a cleft
braided Hopf crossed product, we obtain a simpler mixed complex, that also
gives the Hochschild, cyclic, negative and periodic homology of E.
