Comment #3 on issue 2010 by asmeurer: Integration with the full Risch
Algorithm
http://code.google.com/p/sympy/issues/detail?id=2010
Here we can see the trade-off: heuristic Risch algorithm is simple to
implement and can handle lots of
special cases, but can't prove anything; on contrary Risch algorithm can
prove that a function has (or not)
an elementary integral, but is very complex and hard to implement.
Also, as you can see from the timings above, the full algorithm is *way*
faster than the heuristic. It's easy to see why this is. The full
algorithm integrates directly, but the heurisch tries to build a candidate
integral , which is necessarily quite complex, and solves an often enormous
system of linear equations to solve for the coefficients. Originally, I
was going to use the heurisch as a fast preprocessor to the full algorithm
as Bronstein suggests, but now I think I am going to use it only as a
fallback (because it can handle some algebraic integrals).
Of course, the fact that the full algorithm is using Poly everywhere helps
too :)
I think that adding NonElementaryIntegral class would be sufficient. It
would inherit from Integral class, but
have noop _eval_integral(). There would be also additional classes for
special cases, e.g. logarithmic or sine
integrals (special pretty printing etc.).
Yes, I've though about either adding a subclass NonElementaryIntegral, or
else adding an attribute Integral.has_elementary_integral, which would
True, False, or None (None for when it can't say, such as an algebraic
integrand). Also to consider is that you might want only an elementary
integral, or you might want special functions. So
integrate(2*exp(-x**2)/sqrt(pi), x)
would return erf(x), but
integrate(2*exp(-x**2)/sqrt(pi), x, elementary=True)
would return NonElementaryIntegral(2*exp(-x**2)/sqrt(pi), x). And do you
think NonElementaryIntegral should be pretty-printed any differently, so
that people can recognize it? It seems to me that it should, but maybe
it's just because I'm so excited about the algorithm being able to prove
that integrals are non-elementary.
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