Symbolic Integration Rules

Rubi systematically applies precisely defined rules to efficiently integrate large classes of mathematical expressions. The integration rules in the Rubi source files are expressed in both human and machine-readable form. These files are available for viewing as PDF files and for downloading as Mathematica notebook files. The Rubi Software License grants permission to use the mathematical knowledge in these copyrighted files free of charge.

The integration rules are sequentially assigned a rule number as they are loaded into Rubi. Shown below is the range of numbers of the rules defined in each source file. The Integration Rule Index also lists the range of rule numbers in each file, but arranged in numerical order. To download the rule index as a Mathematica notebook file, click on Integration Rule Index File.

When displaying integration steps, the rule number of each rule applied can be shown as described in the “Displaying integration steps” section of Using Rubi. Using the Integration Rule Index you can determine the source file a rule is defined in. Then you can find the rule’s definition in the file by a simple text search.

The rules defined in the source files are displayed in standard, 2-dimensional mathematical notation, and followed by the equivalent Mathematica program code. Comments preceding many of the rules document them as follows:

The links below to the Rubi source files are organized into categories based on the form of the integrands handled by each file. A zip file containing all the rule files organized in a hierarchical directory structure is also available for downloading by clicking on PDF rule files or Notebook rule files.

1 Algebraic Function Rules

1.1 Binomial products

1.1.1 Linear binomials

1.1.2 Quadratic binomials

1.1.3 General binomials

1.1.4 Improper binomials

1.2 Trinomial products

1.2.1 Quadratic trinomials

1.2.2 Quartic trinomials

1.2.3 General trinomials

1.2.4 Improper trinomials

1.3 Miscellaneous algebraic functions

2 Exponential Rules

3 Logarithm Rules

4 Trig Function Rules

4.1 Sine and cosine

4.3 Tangent and cotangent

4.5 Secant and cosecant

4.7 Miscellaneous trig functions

5 Inverse Trig Function Rules

5.1 Inverse sine and cosine

5.3 Inverse tangent and cotangent

5.5 Inverse secant and cosecant

6 Hyperbolic Function Rules

6.1 Hyperbolic sine and cosine

6.3 Hyperbolic tangent and cotangent

6.5 Hyperbolic secant and cosecant

6.7 Miscellaneous hyperbolic functions

7 Inverse Hyperbolic Function Rules

7.1 Inverse hyperbolic sine

7.2 Inverse hyperbolic cosine

7.3 Inverse hyperbolic tangent and cotangent

7.5 Inverse hyperbolic secant and cosecant

8 Special Function Rules

9 Miscellaneous Rules