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Rubi is a rule-based integrator with a broad set of currently over 6600 integration rules. If repeatedly applying correct transformations. If systematically applied, the integration rules provided here can determine the antiderivative of a wide variety of mathematical expressions. Organized as a decision tree based on the form of the integrand, the 6600+ rules Rubi uses can be viewed in human-readable form or downloaded in machine-readable form.
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.
I would like to challenge the community of Rubi users to find the hardest problem the system can integrate. To that end I will award $1000 (U.S.) to the person who sends me before August 1, 2018, the expression having the largest rule-to-size ratio as displayed by the Int[expn, var, Stats] command with Rubi 4.15 running on Mathematica 11.3. If the largest rule-to-size ratio is a tie, the prize will be awarded to the first entry I had received. Candidate expressions must be formatted in Mathematica syntax and sent to me via email.
The hardest integral received thus far is Int[ArcCoth[x^16]^2,x] which has a rule-to-size ratio of 7.5. Please only send entries having ratios greater than 7.5.
Rubi dramatically out-performs Maple and Mathematica (the two major commercial computer algebra systems) on a grueling integration test suite. Consisting of over 70 thousand integrands and optimal antiderivatives, the entire test suite can be downloaded by clicking on the Test Problems menu option above. This chart shows the percentage of test suite problems for which these systems were able to find optimal antiderivatives. What constitutes an optimal antiderivative is defined following the table below.