Editing Risch algorithm (section)

{{Use mdy dates|date=April 2022}}
{{short description|Method for evaluating indefinite integrals}}
{{calculus|expanded=integral}}
In [[symbolic computation]], the '''Risch algorithm''' is a method of indefinite integration used in some [[computer algebra system]]s to find [[antiderivative]]s. It is named after the American mathematician [[Robert Henry Risch]], a specialist in computer algebra who developed it in 1968.

The [[algorithm]] transforms the problem of [[integration (calculus)|integration]] into a problem in [[differential algebra|algebra]]. It is based on the form of the function being integrated and on methods for integrating [[rational function]]s, [[Nth root|radical]]s, [[logarithm]]s, and [[exponential function]]s. Risch called it a [[decision procedure]], because it is a method for deciding whether a function has an [[elementary function]] as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions.{{Example needed|date=December 2021}}

The complete description of the Risch algorithm takes over 100 pages.<ref>{{harvnb|Geddes|Czapor|Labahn|1992}}.</ref> The '''Risch–Norman algorithm''' is a simpler, faster, but less powerful variant that was developed in 1976 by [[Arthur Norman (computer scientist)|Arthur Norman]].

Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller.<ref>{{Cite web |last=Miller |first=Brian L. |date=May 2012 |title=On the integration of elementary functions: Computing the logarithmic part |url=https://ttu-ir.tdl.org/items/f7a0f000-885f-49f4-a066-77cb9f3fea6b |access-date=2023-12-10}}</ref>