Editing Risch algorithm (section)

==Description==
The Risch algorithm is used to integrate [[elementary function]]s. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations ({{nowrap|+ − × ÷}}). [[Pierre-Simon Laplace|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 {{citation needed|date=June 2021}}. The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s.{{Citation needed|date=November 2021}}

[[Joseph Liouville|Liouville]] formulated the problem that is solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution {{math|''g''}} to the equation {{math|1=''g''′ = ''f''}} then there exist constants {{math|''α<sub>i</sub>''}} and functions {{math|''u<sub>i</sub>''}} and {{math|''v''}} in the field generated by {{math|''f''}} such that the solution is of the form

:<math> g = v + \sum_{i<n} \alpha_i \ln (u_i) </math>

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 the function {{math|''f'' ''e<sup>g</sup>''}}, where {{math|''f''}} and {{math|''g''}} are [[differentiable function]]s, we have

: <math> \left(f \cdot e^g\right)^\prime = \left(f^\prime + f\cdot g^\prime\right) \cdot e^g, \, </math>

so if {{math|''e<sup>g</sup>''}} were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as

: <math> \left(f \cdot(\ln g)^n\right)^\prime =  f^\prime \left(\ln g\right)^n + n f  \frac{g^\prime}{g} \left(\ln g\right)^{n - 1} </math>

then if {{math|(ln ''g'')<sup>''n''</sup>}} were in the result of an integration, then only a few powers of the logarithm should be expected.