  We consider the Hodge filtration on the sheaf of meromorphic functions along
free divisors for which the logarithmic comparison theorem holds. We describe
the Hodge filtration steps as submodules of the order filtration on a cyclic
presentation in terms of a special factor of the Bernstein-Sato polynomial of
the divisor and we conjecture a bound for the generating level of the Hodge
filtration. Finally, we develop an algorithm to compute Hodge ideals of such
divisors and we apply it to some examples.
