  We consider a three dimensional complex polynomial, or rational, vector field
(equivalently, a two-form in three variables) which admits a Liouvillian first
integral. We prove that there exists a first integral whose differential is the
product of a rational 1-form with a Darboux function, or there exists a Darboux
Jacobi multiplier. Moreover, we prove that Liouvillian integrability always
implies the existence of a first integral that is obtained by two successive
integrations from a one-forms with coefficients in a finite algebraic extension
of the rational function field.
