The tan(x/2) substitution, also called the Weierstrass substitution, is one method currently used by computer-algebra systems for the evaluation of trigonometric integrals. The method needs to be improved, because the expressions obtained using it sometimes contain discontinuities, which unnecessarily limit the domains over which the expressions are… (More)
Taking the specific problem domain of indefinite integration, we describe the ongoing development of a repository of mathematical knowledge based on transformation rules. It is important that the repository be not confused with a look-up table. The database of transformation rules is at present encoded in Mathematica, but this is only one convenient form of… (More)
When a computer algebra system has an assumption facility, it is possible to distinguish between integration problems with respect to a real variable, and those with respect to a complex variable. Here, a class of integration problems is deened in which the integrand consists of compositions of continuous functions and signum functions, and integration is… (More)
An algorithm is given for the integration of a class of piecewise-continuous functions. The integration is with respect to a real variable, because the functions considered do not in general allow i n tegration in the complex plane to be deened. The class of integrands includes commonly occurring waveforms, such a s s q u a r e w aves, triangular waves, and… (More)
This paper describes continuing progress on the development of a repository of transformation rules relevant to indefinite integration. The methodology, however, is not restricted to integration. Several optimization goals are being pursued, including achieving the best form for the output, reducing the size of the repository while retaining its scope, and… (More)
We describe the development of a term-rewriting system for indefinite integration; it is also called a rule-based evaluation system. The development is separated into modules, and we describe the module for a wide class of integrands containing the tangent function.
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail. Since they are such a restricted subset of algebraic numbers, it might seem that good simplification of them must already… (More)