---
tag: VJ51B7J125
kind: concept
name: Risch algorithm
msc: [68W30, 12H05, 33F10]
sources:
  - url: https://en.wikipedia.org/wiki/Risch_algorithm
same_as:
  wikidata: Q1382512
  wikipedia: Risch algorithm
---

## Summary

The Risch algorithm is a procedure of symbolic integration.
Given an elementary function, it decides whether the function has an elementary [[antiderivative|0BYWRWFT4F]], and it finds one when it exists.
It rests on [[Liouville's theorem|8VRTPECBPZ]] of differential algebra, which fixes the shape that such an antiderivative must have.
It is named after Robert Henry Risch, and parts of it are implemented in computer algebra systems.

## Explanations

### L1 human

Integration in calculus courses looks like a collection of tricks: substitution, integration by parts, partial fractions.
The Risch algorithm replaces the tricks by a procedure that always ends with an answer.
The answer is either an explicit antiderivative or a proof that none can be written with the usual functions.
For example, it returns $(x - 1) e^x$ for $x e^x$, and it reports that $e^{x^2}$ has no elementary antiderivative.
It views a function as an element of a tower of fields, built from the rational functions by adding logarithms, exponentials and algebraic functions one at a time.