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Suppose, a function $f(x)$ has an elementary antiderivate.

Is it always possible to find the antiderivate with the usual integration mehods (integration by parts, substitution, and so on) or can the Risch algorithm be necessary to find it ?

If I understand it right, the Risch-algorithm is nearly always successful, but I have no idea how the algorithm actually works.

Many integrals (assuming that an elementary antiderivate exists) are solveable with the usual methods as well, but I think there are cases which are too hard, so that we actually need the Risch-algorithm.

Additional question : How efficient is the Risch-algorithm in practice ? Are there cases prcatically infeasible becuase the Risch-agorithm would take too long (Of course, the length of the antiderivate should not be too large) ?

Peter
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1 Answers1

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The Risch integration algorithm is always successful (not "nearly always"), i.e. if an elementary antiderivative exists then it will compute it, and if one does not then it will correctly prove that. The only caveat is that the algorithm depends on the ability to decide equality in the constant subfield (typically some subfield of $\,\Bbb C,\,$ e.g. $\,\Bbb Q[e,\pi,i]),\,$ but equality is generally undecidable for such fields, leading to difficult problems in transcendental number theory, e.g. see Schanuel's conjecture and Richardson's theorem. However, these theoretical results do not typically impinge on the type of problems that arise in practice.

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Bill Dubuque
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