Rule-based Mathematics

Symbolic Integration Rules

Crafted by Albert D. Rich, Applied Logician

If systematically applied, the integration rules provided on this website can determine the antiderivative for a wide variety of mathematical expressions. 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 over 43 thousand integration problems.

The following table summarizes the result of running the integration test suite on these systems as of 8 May 2013:

			Integration Test Suite Results
		Rubi 4				Maple 17				Mathematica 9
Integrand Types	Problems	Optimal	Messy	Unable	Timeout	Optimal	Messy	Unable	Timeout	Optimal	Messy	Unable	Timeout
Algebraic binomial functions	8832	8832	0	0	0	7317	1094	421	0	8275	529	28	0
Algebraic trinomial functions	7013	7013	0	0	0	5320	1559	112	22	6569	425	18	1
Algebraic functions	1635	1624	6	5	0	1274	231	130	0	1416	167	52	0
Exponential functions	838	834	0	4	0	669	55	114	0	696	114	27	1
Logarithm functions	1049	1047	0	2	0	403	176	467	3	922	99	27	1
Trig functions	13183	13140	4	39	0	7657	3936	1554	36	8733	3670	167	613
Inverse trig functions	1612	1609	0	3	0	1122	186	300	4	1406	111	57	38
Hyperbolic functions	3761	3747	1	13	0	2704	488	562	7	3063	573	0	125
Inverse hyperbolic functions	4099	4093	0	6	0	2343	295	1451	10	3726	199	98	76
Special functions	1241	1241	0	0	0	865	56	320	0	986	27	218	10
Contributed problems	200	185	2	13	0	152	15	33	0	155	26	17	2
Totals	43463	43365	13	85	0	29826	8091	5464	82	35947	5940	709	867
Percentages		99.8%	0.0%	0.2%	0.0%	68.6%	18.6%	12.6%	0.2%	82.7%	13.7%	1.6%	2.0%

The following summarizes the meaning of the numbers under the column headings in the above table:

Problems: the number of integration problems for each integrand type.
Optimal: for Rubi 4, the number of results identical to the optimal antiderivative; for the other systems, the number of results no more than twice the size of the optimal antiderivative, based on leaf counts.
Messy: for Rubi 4, the number of results that are correct but not identical to the optimal antiderivative; for the other systems, the number of results that are more than twice the size of the optimal antideriative.
Unable: the number of problems the system returns unintegrated but that are actually integrable in terms of functions known by the system.
Timeout: the number of problems the system fails to integrate within a 25 second timelimit.

Click on Highlights of Integration Test Results to see numerous examples comparing the Rubi, Maple and Mathematica integrators.

To see a table showing how recent versions of Rubi and four other symbolic integrators perform on a test-suite of symbolic integration problems written independently of Rubi, click on one of the following:

Click on A Knowledge Repository for Indefinite Integration Based on Transformation Rules to see an article describing the principles used to the build system of over 5000 rules Rubi uses to efficiently integrate a wide variety of mathematical expressions.

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:

If you have access to Mathematica 7 or better, you can download and run Rubi 4.4 for yourself. In addition to being able to integrate problems like those in the test suite, Rubi provides the option of showing the rules required to integrate expressions, along with the intermediate results. Extract the contents of the Rubi 4.4 zip file into a directory of your choice. Then from Mathematica, open the notebook file Rubi4.4.nb. Click on the sample integration problem at the end of the notebook and press Shift-Enter to evaluate it. After a minute or so depending on the speed of your computer, the first step of the integration should be displayed. To see successive steps, click on the intermediate results and press Shift-Enter.

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.

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.

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