Another idea would be to implement a basic table lookup for such integrals (https://code.google.com/p/sympy/issues/detail?id=1393). This would be the least powerful, as it would only work for expressions exactly of the form we put in the table, but it would be fast, and wouldn't require figuring out how to make thing work with the heurisch or Meijer G algorithms.
Aaron Meurer On Sun, Sep 8, 2013 at 5:46 PM, Aaron Meurer <[email protected]> wrote: > The problem is that sqrt(1/x) is not the same as 1/sqrt(x). If you are > working in a situation where it is, you can do integrate(1/sqrt(1 - > x**2), x) and this works. > > These algebraic integrals are tricky. They only work with the current > heurisch algorithm "by accident", because they happen to come out to > the right form for it to recognize the antiderivative. But since > algebraic functions have multiple equivalent forms that look > completely different to the algorithm (e.g., sqrt(x)/x == 1/sqrt(x)), > and SymPy's default canonicalization is only based on combining > powers, this doesn't happen in general. > > The *right* solution is to implement the algebraic Risch algorithm. > This is a lot of work, though Katja's GSoC project will hopefully > implement some of the needed algorithms in the polys. The workaround > is to modify the heuristics. In heurisch, there is a part that does > some pattern matching and tries to add terms here > https://github.com/sympy/sympy/blob/master/sympy/integrals/heurisch.py#L278. > I suppose we could add a pattern for sqrt(a**2/(a**2 - x**2)) (or > something more general than that). You have to be careful with this, > though, because the more terms you add to the candidate integral, the > more powerful the algorithm becomes, but also, the slower it becomes. > > It might also be possible to modify the Meijer G algorithm to work > with this. I don't know enough about how to do this. I think it means > finding and adding some entry to the lookup table (which is currently > http://docs.sympy.org/dev/modules/integrals/g-functions.html#implemented-g-function-formulae). > Tom and/or Raoul can comment more on this approach. Modifying Meijer G > would be preferable to modifying heurisch if it can work, because the > algorithm is not a slow, and it will also automatically make some > definite integrals work. > > Aaron Meurer > > On Sun, Sep 8, 2013 at 5:14 PM, Ondřej Čertík <[email protected]> wrote: >> Hi, >> >> Currently SymPy can't do this integral: >> >> integrate(sqrt(a**2/(a**2-x**2)), x) >> >> and I reported it here: >> >> https://code.google.com/p/sympy/issues/detail?id=4010 >> >> It seems SymPy cannot even do this integral: >> >> integrate(sqrt(1/(1-x**2)), x) >> >> the solution of which is asin(x), at least for real "x". >> >> Any ideas how to "help" sympy do this kind of integral? >> >> Ondrej >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at http://groups.google.com/group/sympy. >> For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. For more options, visit https://groups.google.com/groups/opt_out.
