Manuscript discussing Weierstrass substitution and Risch algorithm Does anyone know of a manuscript that discusses "Weierstrass substitution"(also called tangent half angle substitution) in Risch algorithm?
There is one paper that discusses the Weierstrass substitution in symbolic integration but "not" specifically in terms of Risch algorithm, this one http://www.apmaths.uwo.ca/~djeffrey/Offprints/toms1994.ps
 A: The Risch integration algorithm works for generalized compositions of algebraic functions, $\exp$ and/or $\ln$ (i.e. in differential field extensions with algebraic, exponential and/or logarithmic elements). It cannot treat trigonometric functions themselves.
A trigonometric function in the integrand can be transformed into its exp/ln-form. But for a rational function of trigonometric functions, there is an easier way. It can be transformed by Weierstrass substitution into a rational function. The antiderivative of this rational function can be found by the Risch algorithm, and subsequent backward substitution gives the antiderivative of the original rational function of trigonometric functions then. Weierstrass substitution is therefore part of an integration algorithm, but it is not part of Risch algorithm.
Neil Langmead: The Weierstrass substitution in REDUCE. Konrad-Zuse-Zentrum für Informationstechnik (ZIB), Berlin, Germany, 1997
http://www.reduce-algebra.com/manual/manualse166.html
https://asmeurersympy.wordpress.com/2010/07/24/the-risch-algorithm-part-2-elementary-functions/
