In Part I of this paper, we givc an extension of Liouville's Theorem and give a nutnber ol examples which show that Integration with special functions involves some phenomena that do not occur in Integration with the elementary functions alone. Our main result generalizes Liouville's Theorem by allowing, in addition to the elementary functions, special functions such as the error Function, Fresnel integrals and the logarithmic integral (but not the dilogorithm or exponential integral) to appear in ihe integral of an elementary funetion. The basic conclusion is that these functions, if they appear, appear linearly. We give an algorithm which decides if an elementary funetion, built up using only exponential functions and rational Operations has an integral which can be expressed in terms of elementary functions and error functions.