There is no algorithm which can find all the cycles in a directed graph in polynomial time. Suppose, the directed graph has n nodes and every pair of the nodes has connections to each other which means you have a complete graph. So any non-empty subset of these n nodes indicates a cycle and there are 2^n-1 number of such subsets. So no polynomial time algorithm exists.
So suppose you have an efficient (non-stupid) algorithm which can tell you the number of directed cycles in a graph, you can first find the strong connected components, then applying your algorithm on these connected components. Since cycles only exist within the components and not between them.
