An extension of Liouville's theorem

I . Motivation, Computing integrals h0s been a favorite pastime of algebraic manipulators (bol, h human and machine) for some time. The usual problem is to determine if the integral of a function can be expressed in terms of some prespecitled set of functions. This set is usually the "elementary functions"-algebraic functions, the exponential function and the logarithm. Slaglc's thesis [11] was the first research to draw the attention of researchers in algebraic manipulation to the integration probtcm. His methods were entircly heuristic and were really an application of artificial intelligence techniques to an algebraic manipulation problem. His program achieved about the same proficiency as that of a college freshman. The first computational work on the integration problem that made usc of thc mathematical structure of the problem is contained in Moses's thesis [4]. SIN, tile program described in Moses's thesis, contained a set of special case algorithms, such as for rational functions of simple exponentials. It also tried as a default method--the "Edge" hcurlstic. This heuristic is used to make an educated guess on the structure of the intcgrat based on Liouvillc 's theorem [2, 3]. By carefully analyzing the mathematical properties of integration, Moses was frequently able to guess the structure of an integral. Then, by filling in the unknown coefficients, differentiating and solving the resulting system of equations, he was able to solve a large class of problems. At the end of the sixties, Risch was able to produce a decision procedure for integrating expressions ccnst.ruct.cd from elementary functions [7, 8]. By sharpening Liouville 's theorem hc was able to deduce what the structure of an integral had to bc if it could bc expressed in terms of elementary functions. The next step in the integration problem is to incorporate special functions in the integral. Moses [5] began discussing the problems and some possible solutions in the use of special functions. Hc emphasized the us,~ge of a "functional" approach to the problem. Ra the r than studying a particular function, Sp(x), Moses suggested studying the s tructure of extensions via Spence functions, Sp(u(x)) for some class of functions u(:c). Rothst.ein investigated the integration problem involving special functions in his thesis [1% Hc developed an algorithm that could compute the integral of an expression given the the functions that could occur in the answer. From our point of view it would have