Symbolic computation Implementation

Question

After using Wolfram Alpha and MATLAB's Symbolic Math Toolbox for solving integrals, ODEs and PDEs, I got curious to know how would I implement an analytical(closed-form) integration(or non-trivial equation) solver.

For example, how could i programmatically solve the following integral analytically?

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Big topic, too big for SO really. But there is a lot of information available on the Internet, I suggest you get your search engine out. — High Performance Mark, Sep 17 '14 at 10:14
Too bad, been having a hard time finding something useful on the matter. — muzio, Sep 17 '14 at 10:21
One of the developers of the `Integrate` function in Mathematica has written an article about this topic. You can read it at http://www.sigsam.org/bulletin/articles/175/lichtblau.pdf — Jommy, Sep 17 '14 at 11:46
What a wonderful citation. Thank you for posting. A great read. — duffymo, Sep 17 '14 at 12:08
One of the first methods mention in that article is the [Risch algorithm](https://en.wikipedia.org/wiki/Risch_algorithm) that might be a good place to start. — Salix alba, Sep 17 '14 at 17:02

score 1 · Answer 1 · answered Sep 17 '14 at 16:35

Integrals are solved by (very complicated) pattern matching. If the integrand looks like the square root of something, then the integral is ...; if it looks like a rational function, then the integral is ..., if exponential, then ...., etc etc etc. There are at least two major difficulties. One is recognizing that an integrand matches one pattern or another, the other is constructing the solution once you have a pattern match. The paper by Lichtblau cited above is about the second part. As to pattern matching on expressions, try a web search for "pattern matching" or "unification". As it happens, pattern matching is most naturally expressed in Lisp, but it can also be handled in other programming languages, usually by reinventing a subset of Lisp.

Symbolic computation Implementation

1 Answers1