Abstract
We were taught at calculus classes that integration is an art, not a science (in contrast to differentiation—even a monkey can be trained to take derivatives). And we were taught wrong. The Risch algorithm (which is known for decades) allows one to find, in a finite number of steps, if a given indefinite integral can be taken in elementary functions, and if so, to calculate it. This algorithm has been constructed in works by an American mathematician Risch near 1970; many cases were not analyzed completely in these works and were later considered by other mathematicians. The algorithm is very complicated, and no computer algebra system implements it fully. Its implementation in Mathematica is rather complete, even with extensions to some classes of special functions, but details are not publicly known. Strictly speaking, it is not quite an algorithm, because it contains algorithmically unsolvable subproblems, such as finding out if a given combination of elementary functions vanishes. But in practice computer algebra systems are quite good in solving such problems. Here we shall consider, at a very elementary level, the main ideas of the Risch algorithm; see [16] for more details.
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Grozin, A. (2014). Risch Algorithm. In: Introduction to Mathematica® for Physicists. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00894-3_10
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