The transcendental Risch differential equation

We present a new rational algorithm for solving Risch differential equations in towers oftranscendental elementary extensions. In contrast to a recent algorithm of Davenport we do not require a progressive reduction of the denominators involved, but use weak normality to obtain a formula for the denominator of a possible solution. Implementation timings show this approach to be faster than a Hermite-like reduction. © 1990, Academic Press Limited. All rights reserved.