Symbolic Integration Rules

Rubi systematically applies precisely defined rules to efficiently integrate large classes of mathematical expressions. The integration rules, expressed in both human and machine-readable form, are available for viewing as PDF files and for downloading as Mathematica notebook files. The file links below are organized into categories based on the form of the integrand.

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.

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. To see the numbers of the rules defined in each of the rule files, click on Integration Rule Index. To download the rule index as a notebook file, click on Integration Rule Index File. Once you know the file a rule is defined in, it is a relatively simple matter to do a text search for it.

The rules in the 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:

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