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I'm looking for an introductory text on algebraic treatment of elementary functions. Really short and easy-going. Video lectures are even better.

I want to learn basic ideas (i.e. definitions) behind integration and differentiation of elementary functions.

I'm also interested in how computer algebra systems like REDUCE or AXIOM treat them as rational functions with some "kernels" (exponents and logarithm's and so on).

Though ideally I'd be happy to start with just polynomials and exponents (sin, cos). So to write symbolic integration and differentiation for a simpler class of functions (no logarithms, division and roots).

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Have a look at e.g. chapter 11 of Geddes/Czapor/Labahn. –  J. M. Apr 20 '12 at 18:31
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The Geddes (Maple) is very good. A tutorial from Bronstein (Axiom) is also available here. –  Raymond Manzoni Apr 28 '12 at 8:35
    
Thank you guys. To that matter there is also "Computer algebra: systems and algorithms for algebraic computation" by Davenport, Siret, Tournier, which is quite common, e.g. one can get it from staff.bath.ac.uk/masjhd/masternew.pdf –  Yrogirg Apr 28 '12 at 16:55

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The answers are provided in the comments. It is a good idea to stick first with books about computer algebra in general, not just about one specific topic.

There are two books to start with

As you see, the second one is free. For symbolic integration it turns out there is a plenty of literature.

MSE question Is there CAS design wisdom regarding categorizing expressions? may help a bit too.

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