The shortest cycle cover conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx1.619m$, and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx1.630m$.