The first part of the paper [J. Symb. Comp. 13, 525-543 (1992; Zbl 0759.12001)] described an algorithm which determines whether an element of a certain class of transcendental Liouvillian functions has an erf- elementary integral, i.e. an integral built up from elementary functions and error-functions, and which calculates it in case of existence. Here this algorithm is extended to a much larger class of functions. Due to a somewhat technical restriction in the definition of the admissible differential fields there are some transcendental Liouvillian functions which have erf-elementary integers but which the algorithm cannot handle; an example of such a function is given.