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  "body": "\n## Summary\n\nThe Risch algorithm is a procedure of symbolic integration.\nGiven an elementary function, it decides whether the function has an elementary [[antiderivative|0BYWRWFT4F]], and it finds one when it exists.\nIt rests on [[Liouville's theorem|8VRTPECBPZ]] of differential algebra, which fixes the shape that such an antiderivative must have.\nIt is named after Robert Henry Risch, and parts of it are implemented in computer algebra systems.\n\n## Explanations\n\n### L1 human\n\nIntegration in calculus courses looks like a collection of tricks: substitution, integration by parts, partial fractions.\nThe Risch algorithm replaces the tricks by a procedure that always ends with an answer.\nThe answer is either an explicit antiderivative or a proof that none can be written with the usual functions.\nFor example, it returns $(x - 1) e^x$ for $x e^x$, and it reports that $e^{x^2}$ has no elementary antiderivative.\nIt views a function as an element of a tower of fields, built from the rational functions by adding logarithms, exponentials and algebraic functions one at a time.\n",
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