Rule-based Mathematics

Indefinite Integration Reduction Rules

Crafted by Albert D. Rich, Applied Logician


If systematically applied, the reduction rules provided on this website can determine the antiderivative for a wide variety of integrands. As proof, a rule-based integrator nicknamed Rubi was implemented using these rules. Rubi dramatically out-performs Maple and Mathematica (the two major commercial computer algebra systems) on a test suite of well over 18 thousand integration problems. The following table summarizes the result of running the test suite on these systems:

Indefinite Integration Test Suite Results
9 June 2011
Rubi 3 Maple 13 Mathematica 7
Integrand Types Problems Optimal Messy Unable Invalid Optimal Messy Unable Invalid Optimal Messy Unable Invalid
Rational functions 1745 1744 0 1 0 1372 350 2 21 1613 123 8 1
Algebraic functions 1906 1890 7 9 0 1187 272 87 360 1763 94 39 10
Exponential functions 592 588 0 4 0   368 67 55 102   548 15 15 14
Logarithm functions 682 679 1 2 0 275 134 236 37 628 32 22 0
Trig functions 8871 8862 4 5 0 2670 5308 217 676 5401 2079 231 1160
Inverse trig functions 559 555 1 3 0 291 126 27 115 509 31 19 0
Hyperbolic functions 2431 2424 4 3 0 766 1120 389 156 2094 293 27 17
Inverse hyperbolic functions 851 848 0 3 0 213 119 307 212 780 56 13 2
Error and Fresnel functions 240 240 0 0 0 164 0 68 8 177 0 63 0
Integral functions 250 250 0 0 0 207 10 28 5 224 6 20 0
Special functions 449 449 0 0 0 188 10 140 111 290 0 159 0
Contributed problems 180 173 5 2 0 122 32 15 11 142 22 13 3
Totals 18756 18702 22 32 0 7823 7548 1571 1814 14169 2751 629 1207
Percentages   99.7% 0.1% 0.2% 0.0% 41.7% 40.2% 8.4% 9.7% 75.5% 14.7% 3.4% 6.4%
Column headings:
Problems: the number of integration problems for each type of integrand.
Optimal: the number of results that are optimal or no more than twice the size of the optimal antiderivative, in terms of leaf counts.
Messy: the number of results that are correct but more than twice the size of the optimal antiderivative.
Unable: the number of problems the system returns unintegrated or cannot integrate in 60 seconds.
Invalid: the number of results that are incorrect or the system is incapable of verifying correct by differentiation.

Highlights of the Indefinite Integration Test Results gives numerous eye-opening comparisons of the Rubi, Maple and Mathematica integrators.

A Knowledge Repository for Indefinite Integration Based on Transformation Rules describes the principles used to build this repository.

To view or download the rules Rubi uses to integrate expressions, click on one of the following file types:
To view or download the indefinite integration problems in the test suite, click on one of the following formats:
To view or download the raw indefinite integration test results as generated by these systems, click on one of the following:
I encourage the submission of new rules and test problems, preferably in the same format as the files on this website. Please send your comments and suggestions to Albert_Rich@msn.com

The mathematical knowledge on this website is freely available for any educational, academic or commercial use. Please include the website address and appropriately acknowledge its author in any product incorporating its contents.

free counters

Maple is a registered trademark of Maplesoft.
Mathematica is a registered trademark of Wolfram Research, Inc. who generously provided a copy of Mathematica 8.0.1 to support this research.