Welcome to Rubi, the Rule-based Integrator

By systematically applying its extensive, coherent collection of symbolic integration rules, Rubi is able to find the optimal antiderivative of large classes of mathematical expressions. Also Rubi can show the rules and intermediate steps it uses to integrate an expression, making the system a great tool for both learning and doing mathematics.

The over 6700 rules Rubi uses are organized as a decision tree based on the form of the integrand. Click on Rules to view the mathematical formula and application conditions for each rule expressed in standard 2-D mathematical notation.

If the Mathematica computer algebra system is available on your computer, click on Install Rubi for instructions how to download and install the system. Then click on Using Rubi to learn how to integrate expressions and optionally show the steps required to do so.

An extensive test-suite of over 72,000 integration problems is used to test each new release of Rubi. Since it can also be used to test other symbolic integrators, the test-suite has been translated into the syntax of several popular computer algebra systems. Click on Test Problem to download the available translations of the test-suite. Click on Test Results to view the very illuminating results of running the test-suite on Rubi and on Mathematica’s built-in integrator.

Click on Vision to read a short, plain-language essay describing the numerous advantages of organizing mathematical knowledge as a rule-based decision tree like Rubi does. Finally, click on About to read about Rubi’s development and developers.

Comparison with other symbolic integrators

Rubi dramatically out-performs other symbolic integrators, including Maple and Mathematica, on a grueling test-suite of integration problems. Each problem consists of an integrand and its optimal antiderivative. For each problem the result produced by a system is given a grade as follows:

Nasser M. Abbasi has independently done extensive testing of the integration capabilities of several computer algebra systems. Detailed test results are available at his Computer algebra independent integration tests website. The following chart based on his test results summarizes the grades given Rubi, Mathematica and Maple on the integration test-suite:

integrationTestResultsChart

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