Let \(G\) be a bridgeless graph with \(n\) vertices and \(m\) edges and let \(r\) be the minimum length of an even cycle in \(G\) of length at least 6 \((r=\infty\) if there is no such cycle). It is proved that the edges of \(G\) can be covered by cycles whose total length is at most \(m+(n-1)r/(r- 1)\). The proof suggests a polynomial time algorithm for constructing such a cover.