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If you take a look at WolframAlpha, or other computer algebraic system, you will find that it is able to do symbolic manipulation like real humans.

For example, if you type in an integral, it can show you step by step on how to solve the integration.

What are the algorithms behind all this?

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up vote 9 down vote accepted

The algorithms behind symbolic integration (due to Liouville, Ritt, Risch, Bronstein et al.) are discussed in prior questions here, e.g. the transcendental case and algebraic case.

For general references on symbolic computation see any of the standard texbooks, e.g. Geddes et al. Algorithms for computer algebra, Grabmeier et al: Computer algebra handbook,von zur Gathen: Modern computer algebra, and Zippel: Effective polynomial computation, and many other books. See also the Journal of Symbolic Computation and various conferences: SIGSAM ISSAC, EUROCAL, etc.

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Don't forget A=B, which can be downloaded for free at math.upenn.edu/~wilf/AeqB.html. –  Yuval Filmus Feb 10 '11 at 5:08
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@Yuval: I didn't mention books devoted to specific topics since the OP didn't mention their specific interests (besides integration). Note that my list of general textbooks is far form complete - it's off the top of my head, late at night (and probably biased since I was a Macsyma developer). –  Bill Dubuque Feb 10 '11 at 5:13
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