I know the Risch algorithm fairly well.

I made two screencast videos describing how to use Axiom or Sage to 
simplify one of the integrals used in the Facebook paper.

Quick summary - Axiom works quite well.  Sage can't do it in one step, but 
the new function field features in Sage 9 allow the integral to computed, 
but you have to know something about the Risch algorithm to step through it.

The second video is a whirlwind overview of the Risch theorem and shows how 
to do a Risch calculation (on the Facebook integral) using Sage.

You can view the videos here:

https://www.freesoft.org/blogs/soapbox/the-facebook-integral/


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