Sage uses Pynac/Ginac for its symbolic expression core, which is much 
> faster than Sympy. For more complicated symbolic computations like 
> integration we use some third-party programs. My guess would be that maxima 
> is better at integrating than Sympy, which is why its the default here. 
> There is nothing wrong with Sympy, and we should use it wherever it is to 
> our advantage.
>
>
+1

>From what I have seen, there are some integrations and sums and whatnot 
where Maxima is better, some Sympy.  They have some nice recurrence stuff, 
as I believe you've seen.  My personal desire would be for Sage to try both 
for any given integration (maybe in the order Maxima, then Sympy) but I 
think that would probably entail too much overhead.
 

> where apparently not much thought was given to conversion of symbolic 
>> functions from SymPy back to Sage.
>> http://trac.sagemath.org/ticket/16801
>>
>  
This is actually true about Maxima functions as well, but because Maxima is 
the default we have spent more time fixing that problem.  Helping with this 
would be very good; Sympy and Sage have complementary goals, so there is no 
reason not to keep good with them - also because it is quite likely we will 
continue to be able to report bugs upstream, which we also do with Maxima.

As for staying away from tickets, naturally if you have a pick it is nice 
if you can pick tickets that will benefit the most people :) but 
open-source development is also largely about scratching itches so if you 
are motivated to do better Sympy translation stuff, that would be awesome; 
it can only help.  The only tickets I would stay away from are ones that 
are clearly fixed in the meantime, in which case you shouldn't stay away 
from those either but instead provide a doctest and set to "needs review" :)

- kcrisman

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