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Integration Test Problems

Contemporaneous with the development of Rubi, a test-suite currently consisting of over 72,000 integration problems has been built to test every aspect of the system. The test-suite files are organized into categories based on the form of the problems' integrands. Each problem is stored as a list of the integrand, the integration variable, the number of steps Rubi uses to integrate the problem, and the optimal antiderivative.

The Integration Test-suites

Since the resulting test-suite is useful for evaluating the performance of any symbolic integrator, the problems have been translated into the syntax of several popular computer algebra systems. The integration test-suite is freely available as a zip file in the following formats:

Note that the optimal antiderivatives (i.e. the correct answers) in the test-suite are valid for all real and complex values of their integration and other variables. Therefore, when the test-suite is used to evaluate a system's symbolic integrator, all the variables in the problem should be declared complex if not by default. For example, in Maxima variables are assumed to be real by default.

Installing the Integration Test Program

You can click on the Test Results menu option to see the results of running Rubi, Mathematica and Maple on the integration test-suite. However, if you have access to Mathematica or Maple and want to run the test-suite yourself, you can download and install the integration test program used to create these test results.

Mathematica version 7 or later is required to run the test program. However, the installation procedure is a little simpler for more recent versions of Mathematica. Instructions for installing and uninstalling the test program for Mathematica and Maple are as follows:

Mathematica 11.2 or later

Start Mathematica and issue the PacletInstall command

PacletInstall["https://rulebasedintegration.org/IntegrationTestProgram-2018.12.22.paclet"]

This downloads the latest version of the test program and installs it on your computer as a package in Mathematica's Paclets repository for packages.

To remove a paclet-based installation of the test program, issue the Mathematica command

PacletUninstall["IntegrationTestProgram"]

Mathematica 9 or later

Click on https://rulebasedintegration.org/IntegrationTestProgram-2018.12.22.paclet to download the test program paclet file to the Downloads directory on your computer. Then start Mathematica and issue the PacletInstall command

PacletInstall[FileNameJoin[{path, "IntegrationTestProgram-2018.12.22.paclet"}]

where path is the full path name of the Downloads directory on your computer. This installs the latest version of the test program on your computer as a package in Mathematica's Paclets repository for packages.

Mathematica 7 or later

Click on https://rulebasedintegration.org/IntegrationTestProgram-2018.12.22.zip to download the test program zip file for Mathematica to the Downloads directory on your computer. Then extract from the zip file the IntegrationTestProgram folder with all its contents into the directory returned by the Mathematica commmand

FileNameJoin[{$UserBaseDirectory, "Applications"}]

This installs the latest version of the test program on your computer in Mathematica's Applications repository for packages.

To remove the test program package for Mathematica, just delete the IntegrationTestProgram folder from the "Applications" directory.

Maple

Click on https://rulebasedintegration.org/MapleIntegrationTestProgram.zip to download the test program zip file for Maple to the Downloads directory on your computer. Then extract from the zip file the IntegrationTestProgram folder with all its contents into a directory of your choice.

Comments at the beginning of the file IntegrationTestProgram.mpl describe how to run the test program and the effect of the control variables.

To remove the test program for Maple, just delete the IntegrationTestProgram folder.

Running the Integration Test Program

After the test program has been installed, load it using the Mathematica's Get command

Get["IntegrationTestProgram`"]

This defines the command functions TestRubi and TestMathematica that test Rubi and Mathematica's built-in integrator, respectively. Also defined are several control variables including $PercentToTest and $TestOnlyElementary. Usage help for these functions and control variables is available. For example,

? TestRubi

summarizes the significance of TestRubi's arguments. The following are the top-level directory names of the test-suites that are available:

  • 0 Independent test suites
  • 1 Algebraic functions
  • 2 Exponentials
  • 3 Logarithms
  • 4 Trig functions
  • 5 Inverse trig functions
  • 6 Hyperbolic functions
  • 7 Inverse hyperbolic functions
  • 8 Special functions

So to test Rubi and Mathematica on collections of integration problems created independently of Rubi's developers, Load Rubi and the test program. Then issue the commands

TestRubi["0 Independent test suites"]
TestMathematica["0 Independent test suites"]

This will create two notebooks showing the problems for which Rubi and Mathematica are unable to produce optimal antiderivatives. Click on the Test Results menu option to see how deficient results are displayed in these notebooks.

If you have several hours of computer time available to test the over 72,000 problems in the test-suite, issue the command

TestRubi[""]

If you have several days of computer time available, issue the command

TestMathematica[""]

Have fun.