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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 than the canonical one, that gives the Hochschild, cyclic, negative and periodic homologies of a crossed product E := A# f V, in the sense of Brzezi´nski.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 homologies 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 homologies of E.