D R A F T - Work in progress

Integrator Test Program

A Mathematica program for testing Rubi and Mathematica's built-in integrator

This Mathematica package consists of an integrator test program and a test suite of over 70000 integration problems. The program can test both Rubi and Mathematica's built-in integrator on all or some of the problems. It generates a notebook showing the problems on which the integrator being tested is deficient. These test results can optionally be saved as a notebook file. Mathematica 7 or later is required to run the test program.

The integrator test program uses test-cases that can be found in MathematicaSyntaxTestSuite. Each Mathematica file contains a set of test-problems which have the following form

{x^(5/2), x, 1, 2*x^(7/2)/7}

The elements of the list have the following meaning:

1. Input expression
2. Integration variable
3. Number of steps necessary to solve the integration
4. Optimal (or best known) antiderivative for the input expression

Download and Install Integrator Test Program

For Mathematica 11.2 or later, use the PacletInstall command

PacletInstall[
  "https://github.com/RuleBasedIntegration/IntegratorTestProgram/releases/download/???/IntegratorTestProgram-???.paclet"
]

to download the current integrator test program paclet directly from the online repository on GitHub, and then install it on your computer as a Mathematica package named IntegratorTestProgram.

Integrator Test Commands

After installing the package, use the Get command

<<IntegratorTestProgram`

to load the test program into Mathematica. This defines the command functions TestRubi and TestMathematica that test Rubi and Mathematica's built-in integrator, respectively.

If filenames is a test file name or list of file names, TestRubi[filenames] tests Rubi on the integration problems in the list of files. For example, the command

TestRubi[$IndependentTestFiles]

tests Rubi on collections of integration problems developed independently by authors of calculus textbooks, implementors of computer algebra systems and others. Running the test generates a notebook that shows the problems on which Rubi is deficient. For each deficient result the following is shown:

• The problem number and a brief explanation why the result is deficient
• The problem integral
• The optimal antiderivative, its expression type, leaf size, and the number of steps Rubi uses to integrate it
• The result produced by the integrator being tested, its expression type, leaf size, and the number of steps used to integrate it (if Rubi is being tested)

If testname is a string, TestRubi[filenames,testname] generates a notebook of deficient test results and saves it as a file with testname and Rubi's version number included in its name. For example, the command

TestRubi[$IndependentTestFiles, "Independent"]

saves the test results as the notebook Rubi 4.16.0.4 Independent Integration Test Results.nb.

The arguments and effect of TestMathematica commands is the same as for TestRubi.

Program Control Variables

The control variable $PercentToTest determines the percentage of problems to test in each file. For example, after making the assignment

$PercentToTest = 20

the TestRubi and TestMathematica commands will randomly select 20% of the problems to test in each file. Thus, reducing by a factor of 5 the time required to complete the test. The default value of $PercentToTest is 100.

The control variable $TimeOutLimit determines the time limit in seconds allowed to integrate a problem. For example, the assignment

$TimeOutLimit = 200

increases the time limit to 200 seconds from the default value of 120 seconds.

Expression Type Numbers

The test results include the expression type number of the optimal antiderivative and of the result produced by the integrator being tested. The expression type number indicates the highest level of functions the expression involves as follows:

• 1 = rational function
• 2 = algebraic function
• 3 = elementary function
• 4 = special function
• 5 = hyperpergeometric function
• 6 = appell function
• 7 = rootsum function
• 8 = integrate function
• 9 = unknown function

When comparing antiderivatives, the test program considers the one having a smaller expression type number to be simpler and therefore superior.