Computer algebra and numerical integration

Algebraic manipulation systems such as MACSYMA include algorithms and heuristic procedures for indefinite and definite integration, yet these system facilities are not as generally useful as might be thought. Most isolated definite integration problems are more efficiently tackled with numerical programs. Unfortunately, the answers obtained are sometimes incorrect, in spite of assurances of accuracy; furthermore, large classes of problems can sometimes be solved more rapidly by preliminary algebraic transformations. In this paper we indicate various directions for improving the usefulness of integration programs given closed form integrands, via algebraic manipulation techniques. These include expansions in partial fractions or Taylor series, detection and removal of singularities and symmetries, and various approximation techniques for troublesome problems.