The Risch algorithm is an algorithm for the calculus operation of indefinite integration (i.e. finding antiderivatives). The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, logarithms, and exponential functions. Robert Risch called the algorithm a decision procedure, because it is a method for deciding if a function has a simple-looking function as an indefinite integral; and also, if it does, determining it. Flowcharts are often used to represent algorithms. ... For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... Wikipedia does not yet have an article with this exact name. ... In mathematics, a rational function is a ratio of polynomials. ... In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. ... The exponential function is one of the most important functions in mathematics. ... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using...

The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, and the four operations (+ − × ÷). Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions. The algorithm suggested by Laplace is usually described in calculus textbooks but was only implemented in the 1960s. In mathematics, several functions are important enough to deserve their own name. ... Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ... Events and trends The 1960s was a turbulent decade of change around the world. ...

Liouville formulated the problem solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution f to the equation g ′ = f then for constants α_i and elementary functions u_i and v the solution is of the form Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ...

Risch developed a method for finding a finite set of elementary functions to consider.

The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. If there is a function f e^g where f and g are functions of x, then

so if e^g were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as

then if lnⁿg were in the result of an integration, then only a few powers of the logarithm should be expected.

The Risch decision procedure is not formally an algorithm because it requires an oracle that decides whether a constant expression is zero, a problem shown by Daniel Richardson to be undecidable. Transforming the Risch decision procedure into an algorithm that can be executed by a computer is a complex task that requires the use of heuristics and many refinements. In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. ...

References

• R. H. Risch (1969). "The Problem of Integration in Finite Terms". Transactions of the American Mathematical Society, 139, 167-189.
• Maxwell Rosenlicht (1972). "Integration in finite terms". American Mathematical Monthly, 79, 963-972.