It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.
Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L.
The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s.
Liouville proved by analytical means that if there is an elementary solution g to the equation g′ = f then there exist constants αi and functions ui and v in the field generated by f such that the solution is of the form Risch developed a method that allows one to consider only a finite set of functions of Liouville's form.
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation.
For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993[3]) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral):[4][5] namely: But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions,[6] as FriCAS also shows.
Some computer algebra systems may here return an antiderivative in terms of non-elementary functions (i.e. elliptic integrals), which are outside the scope of the Risch algorithm.
(SymPy can solve it while FriCAS fails with "implementation incomplete (constant residues)" error in Risch algorithm): Some Davenport "theorems"[definition needed] are still being clarified.
The case of the purely transcendental functions (which do not involve roots of polynomials) is relatively easy and was implemented early in most computer algebra systems.
[10] The case of purely algebraic functions was partially solved and implemented in Reduce by James H. Davenport – for simplicity it could only deal with square roots and repeated square roots and not general radicals or other non-quadratic algebraic relations between variables.
Gaussian elimination will produce incorrect results if it cannot correctly determine whether a pivot is identically zero[citation needed].