Let $A$ be a semi-free commutative differential graded algebra in non-negative degrees with differential $d$ of degree $-1$, over a field of characteristic zero. Recall that semi-free means that if you forget the differential, then $A$ is free as a graded commutative algebra.

In Loday's book on cyclic homology, there is an explanation of how to compute Hochschild and cyclic homology of $A$ using a mixed complex built from a de Rham type complex for $A$ built from K\"ahler differentials and alternating powers of K\"ahler differentials. Is there a similar way to compute negative and periodic cyclic homology? The most naive thing to do is just to use the given mixed complex in the usual way, but that looks very large and it is not clear to me what the output would be.

A reference to such a computation would be very welcome.