Editing Integral (section)

===Symbolic===
{{Main|Symbolic integration}}

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive [[Lists of integrals|tables of integrals]] have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to [[computer algebra system]]s that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like [[Macsyma]] and [[Maple (software)|Maple]].

A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in [[Closed-form expression|closed form]] involving only [[elementary function]]s, include [[rational function|rational]] and [[exponential function|exponential]] functions, [[logarithm]], [[trigonometric functions]] and [[inverse trigonometric functions]], and the operations of multiplication and composition. The [[Risch algorithm]] provides a general criterion to determine whether the antiderivative of an elementary function is elementary and to compute the integral if is elementary. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in [[Mathematica]], [[Maple (software)|Maple]] and other [[computer algebra system]]s, does just that for functions and antiderivatives built from rational functions, [[Nth root|radicals]], logarithm, and exponential functions.

Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the [[special functions]] (like the [[Legendre function]]s, the [[hypergeometric function]], the [[gamma function]], the [[incomplete gamma function]] and so on). Extending Risch's algorithm to include such functions is possible but challenging and has been an active research subject.

More recently a new approach has emerged, using [[D-finite function| ''D''-finite functions]], which are the solutions of [[linear differential equation]]s with polynomial coefficients. Most of the elementary and special functions are ''D''-finite, and the integral of a ''D''-finite function is also a ''D''-finite function. This provides an algorithm to express the antiderivative of a ''D''-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a ''D''-function as the sum of a series given by the first coefficients and provides an algorithm to compute any coefficient.

Rule-based integration systems facilitate integration.  Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands.  This system uses over 6600 integration rules to compute integrals.{{sfn|Rich|Scheibe|Abbasi|2018}}  The [[Ramanujan's master theorem#Bracket integration method| method of brackets]] is a generalization of Ramanujan's master theorem that can be applied to a wide range of univariate and multivariate integrals.  A set of rules are applied to the coefficients and exponential terms of the integrand's power series expansion to determine the integral.  The method is closely related to the [[Mellin transform]].{{sfn|Gonzalez|Jiu|Moll|2020}}