| Case
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Algorithm
|
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Indefinite integrals expressible in terms of elementary functions
|
An extended version of the Risch algorithm is used whenever both the
integrand and the integral can be expressed in terms of elementary
functions.
|
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Other indefinite integrals
|
Heuristic simplification followed by pattern matching is used. The
algorithms in Mathematica CalcCenter cover all of the
indefinite
integrals resulting in elementary functions that are found in standard reference
books such as Gradshteyn-Ryzhik.
|
|
Definite integrals that involve no singularities
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These are mostly done by taking limits of the indefinite integrals.
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Other definite integrals
|
Marichev-Adamchik Mellin transform methods are used. The results are often
initially expressed in terms of Meijer G functions, which are
converted into hypergeometric functions using Slater's Theorem and
then simplified.
|
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Numeric definite integration
|
Adaptive Gaussian quadrature with error estimation based on evaluation
at Kronrod points is used.
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Multidimensional definite integration
|
Adaptive Genz-Malik algorithm is used.
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