I am not sure I will have time to be a mentor this year, but I would like 
to propose porting rubi to sage:

http://www.apmaths.uwo.ca/~arich/

It is a series of rules (over 6000) to be applied to symbolic expressions 
in order to get their primitive. The results they produce are better than 
the Mathematica builtin integrator, they succeed in a bigger number of 
cases, and they also hapen to be faster. If we could include that in Sage 
that would be a killer feature.

Right now the rules are written in Mathematica language, and translating 
them by hand would be too much work for a person to do in a few months. So 
likely the best approach is to try and write a parser that could do it 
automatically. 

I have talked to one of the authors in person during a conference, and he 
really liked the idea of seeing it ported to a free system. In fact I thing 
that there have already been some discussion about this in maxima and sympy 
(maybe we could cooperate in this effort).

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