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What's the hardest integral Mathematica running Rubi can find?

Rubi (Rule-based integrator) is an open source program written in Mathematica's powerful pattern-matching language. The recently released version 4.15 of Rubi at http://www.apmaths.uwo.ca/~arich/ requires Mathematica 7 or better to run. Among other improvements, Rubi 4.15 enhances the functionality of its integrate command as follows:

  • Int[expn, var] returns the antiderivative (indefinite integral) of expn with respect to var.
  • Int[expn, var, Step] displays the first step used to integrate expn with respect to var, and returns the intermediate result.
  • Int[expn, var, Steps] displays all the steps used to integrate expn with respect to var, and returns the antiderivative.
  • Int[expn, var, Stats], before returning the antiderivative of expn with respect to var, displays a list of statistics of the form {a, b, c, d, e} where a) is the number of steps used to integrate expn, b) is the number of distinct rules used to integrate expn, c) is the leaf count size of expn, d) is the leaf count size of the antiderivative of expn, and e) is the rule-to-size ratio of the integration (i.e. the quotient of elements b and c).

The last element of the list of statistics displayed by Rubi's Int[expn, var, Stats] command is the number of distinct rules required to integrate expn divided by the size of expn. This rule-to-size ratio provides a normalized measure of the amount of mathematical knowledge Rubi uses to integrate expressions. In other words, this ratio can be used as a metric showing the difficulty of solving indefinite integration problems. For example, the hardest problem in Rubi's 70,000+ test suite is integrating (a+b ArcTanh[c/x^2])^2 which has a rule-to-size ratio of 2.5.

On Rubi's website are the terms of a challenge, for which there is a substantial prize, for the user who finds the hardest problem Rubi can integrate.

POSTED BY: Albert Rich
POSTED BY: Patrick Scheibe
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