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Extending Integrate[]

• To: mathgroup at smc.vnet.net
• Subject: [mg87754] Extending Integrate[]
• From: Szabolcs Horvát <szhorvat at gmail.com>
• Date: Wed, 16 Apr 2008 06:51:42 -0400 (EDT)
• Organization: University of Bergen

According to the documentation it is possible to extend Integrate[] with 
new rules:

http://reference.wolfram.com/mathematica/tutorial/IntegralsThatCanAndCannotBeDone.html

The specific example that is given is:

Unprotect[Integrate]

Integrate[Sin[Sin[a_. + b_. x_]], x_] := Jones[a, x]/b

Now Integrate[Sin[Sin[3x]], x] will return Jones[0, x]/3

But if I try an integrand that is just slightly more complicated, 
Mathematica returns it unevaluated:

In[21]:= Integrate[a + Sin[Sin[x]], x]
Out[21]= Integrate[a + Sin[Sin[x]], x]

My question is:  Is it really possible to extend Integrate in an 
intelligent way?  The above example seems to be just simple pattern 
matching.  It would work with any function, not just Integrate[].  There 
is nothing new or surprising about it.

But is it *really* possible to extend Integrate[] with new definitions 
in a way that Mathematica will use these rules when calculating more 
complicated Integrals?

If (as I suspect) it is not possible to do this, then the documentation 
is a bit misleading ...