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Is Risch's algorithm powerful enough to determine any integral of a function have a closed form or not?

Is there a historic piece of reference that support your answer?

http://mathworld.wolfram.com/RischAlgorithm.html

Victor
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    http://math.stackexchange.com/q/287442/462 should give you more than enough references. And yes, the algorithm applies in every case, and several references in that list provide complete proofs. – Andrés E. Caicedo May 05 '13 at 18:23

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No, the Risch algorithm doesn't work for all closed-form functions. A closed-form function is a function from a given set of functions where the functions themselves or the generator of the set is given.

The Risch algorithm gives answers only for integrands and antiderivatives from the same differential field, e.g. for the Elementary functions or the Liouvillian functions. Such functions generated by a differential field are defined e.g. in section 1 of Davenport, J. H.: What Might "Understand a Function" Mean. In: Kauers, M.; Kerber, M., Miner, R.; Windsteiger, W.: Towards Mechanized Mathematical Assistants. Springer, Berlin/Heidelberg, 2007, page 55-65. And for such classes of functions, the Risch algorithm is complete. The Risch algorithm uses Liouville's theorem in differential algebra.

There are extensions to the Risch algorithm to be able to handle some special functions.

See e.g.: Wikipedia: Risch algorithm and MathWorld: Risch Algorithm.

IV_
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