The problem of integration in finite terms for elementary functions has been solved since 1969 with the invention of the Risch algorithm. However, ever since then the sine and cosine functions have been rewritten in terms of other functions, originally using complex exponentials. Later, for the Risch-Norman algorithm, an alternative has been proposed where they are rewritten in terms of a tangent of half the angle.

We present extensions to the Risch-Norman algorithm that admit functions satisfying systems of differential equations (and thus also functions satisfying a differential equation of higher order). We further improve the method to allow algebraic relations to exist among the functions, paying particular attention to new logarithms that may appear and need to be predicted. This results in a heuristic but quite powerful algorithm that is able to deal with a large class of special functions and a variety of algebraic functions. In particular, it is able to work with the sine and cosine functions directly without the need to rewrite them in terms of other functions.