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:

Derivation gives the integration technique used to derive a rule.
Basis gives the mathematical identity used to transform the integrand into a form easier to integrate.
Reference gives the number assigned to the rule in one or more of the following integration tables:
- G&R is the “Table of Integrals, Series, and Products”, fifth edition, I.S. Gradshteyn, I.M. Ryzhik, and Alan Jeffrey, editors.
- CRC is the “CRC Standard Mathematical Tables and Formulae”, 29th edition, William H. Beyer, editor.
- A&S is the “Handbook of Mathematical Functions”, Milton Abramowitz and Irene A. Stegun, editors.

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

PDF file Notebook file Rules 0029-0033: 1.1.1.1 (a+b x)^m
PDF file Notebook file Rules 0034-0073: 1.1.1.2 (a+b x)^m (c+d x)^n
PDF file Notebook file Rules 0074-0176: 1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p
PDF file Notebook file Rules 0147-0194: 1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q
PDF file Notebook file Rules 1630-1637: 1.1.1.5 P(x) (a+b x)^m (c+d x)^n
PDF file Notebook file Rules 1623-1629: 1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p
PDF file Notebook file Rules 1596-1622: 1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q

1.1.2 Quadratic binomials

PDF file Notebook file Rules 1824-1829: 1.1.2.x P(x) (a+b x^2)^p
PDF file Notebook file Rules 1813-1823: 1.1.2.y P(x) (c x)^m (a+b x^2)^p

1.1.3 General binomials

PDF file Notebook file Rules 0195-0264: 1.1.3.1 (a+b x^n)^p
PDF file Notebook file Rules 0265-0379: 1.1.3.2 (c x)^m (a+b x^n)^p
PDF file Notebook file Rules 0380-0446: 1.1.3.3 (a+b x^n)^p (c+d x^n)^q
PDF file Notebook file Rules 0447-0534: 1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q
PDF file Notebook file Rules 0535-0580: 1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r
PDF file Notebook file Rules 0581-0620: 1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r
PDF file Notebook file Rules 1864-1914: 1.1.3.7 P(x) (a+b x^n)^p
PDF file Notebook file Rules 1830-1863: 1.1.3.8 P(x) (c x)^m (a+b x^n)^p

1.1.4 Improper binomials

PDF file Notebook file Rules 2025-2037: 1.1.4.1 (a x^j+b x^n)^p
PDF file Notebook file Rules 2038-2058: 1.1.4.2 (c x)^m (a x^j+b x^n)^p
PDF file Notebook file Rules 2059-2067: 1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q
PDF file Notebook file Rules 2068-2078: 1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p

1.2 Trinomial products

1.2.1 Quadratic trinomials

PDF file Notebook file Rules 0621-0639: 1.2.1.1 (a+b x+c x^2)^p
PDF file Notebook file Rules 0640-0776: 1.2.1.2 (d+e x)^m (a+b x+c x^2)^p
PDF file Notebook file Rules 0777-0858: 1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p
PDF file Notebook file Rules 0859-0980: 1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p
PDF file Notebook file Rules 0981-1011: 1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q
PDF file Notebook file Rules 1012-1059: 1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q
PDF file Notebook file Rules 1060-1104: 1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2)
PDF file Notebook file Rules 1671-1675: 1.2.1.8 P(x) (a+b x+c x^2)^p
PDF file Notebook file Rules 1638-1670: 1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p

1.2.2 Quartic trinomials

PDF file Notebook file Rules 1102-1120: 1.2.2.1 (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1121-1156: 1.2.2.2 (d x)^m (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1157-1256: 1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1257-1353: 1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1685-1694: 1.2.2.5 P(x) (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1676-1684: 1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1695-1737: 1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p
PDF file Notebook file Rules 1738-1758: 1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p

1.2.3 General trinomials

PDF file Notebook file Rules 1354-1365: 1.2.3.1 (a+b x^n+c x^(2 n))^p
PDF file Notebook file Rules 1366-1404: 1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p
PDF file Notebook file Rules 1405-1475: 1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p
PDF file Notebook file Rules 1476-1595: 1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p
PDF file Notebook file Rules 1788-1812: 1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p
PDF file Notebook file Rules 1759-1787: 1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p

1.2.4 Improper trinomials

PDF file Notebook file Rules 1915-1924: 1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p
PDF file Notebook file Rules 1925-1945: 1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p
PDF file Notebook file Rules 1946-1953: 1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p
PDF file Notebook file Rules 1954-1969: 1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p

1.3 Miscellaneous algebraic functions

PDF file Notebook file Rules 2079-2095: 1.3.1 P(x)^p
PDF file Notebook file Rules 2096-2127: 1.3.2 P(x) Q(x)^p
PDF file Notebook file Rules 2128-2187: 1.3.3 Miscellaneous algebraic functions
PDF file Notebook file Rules 1970-2024: 1.3.4 Normalizing algebraic functions

2 Exponential Rules

PDF file Notebook file Rules 2027-2220: 2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p
PDF file Notebook file Rules 2221-2224: 2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p
PDF file Notebook file Rules 2225-2331: 2.3 Miscellaneous exponentials

3 Logarithm Rules

PDF file Notebook file Rules 2332-2337: 3.1.1 (a+b log(c x^n))^p
PDF file Notebook file Rules 2338-2349: 3.1.2 (d x)^m (a+b log(c x^n))^p
PDF file Notebook file Rules 2350-2369: 3.1.3 (d+e x^r)^q (a+b log(c x^n))^p
PDF file Notebook file Rules 2370-2398: 3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p
PDF file Notebook file Rules 2399-2435: 3.1.5 u (a+b log(c x^n))^p
PDF file Notebook file Rules 2535-2558: 3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p
PDF file Notebook file Rules 2559-2577: 3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p
PDF file Notebook file Rules 2578-2597: 3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s
PDF file Notebook file Rules 2436-2496: 3.3 u (a+b log(c (d+e x)^n))^p
PDF file Notebook file Rules 2497-2534: 3.4 u (a+b log(c (d+e x^m)^n))^p
PDF file Notebook file Rules 2598-2641: 3.5 Miscellaneous logarithms

4 Trig Function Rules

4.1 Sine and cosine

PDF file Notebook file Rules 2642-2668: 4.1.0.1 (a sin)^m (b trg)^n
PDF file Notebook file Rules 2669-2698: 4.1.0.2 (a trg)^m (b tan)^n
PDF file Notebook file Rules 2699-2712: 4.1.0.3 (a csc)^m (b sec)^n
PDF file Notebook file Rules 2713-2745: 4.1.1.1 (a+b sin)^n
PDF file Notebook file Rules 2746-2784: 4.1.1.2 (g cos)^p (a+b sin)^m
PDF file Notebook file Rules 2785-2812: 4.1.1.3 (g tan)^p (a+b sin)^m
PDF file Notebook file Rules 2813-2911: 4.1.2.1 (a+b sin)^m (c+d sin)^n
PDF file Notebook file Rules 2912-3006: 4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n
PDF file Notebook file Rules 3007-3044: 4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n
PDF file Notebook file Rules 3045-3088: 4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin)
PDF file Notebook file Rules 3089-3107: 4.1.4.1 (a+b sin)^m (A+B sin+C sin^2)
PDF file Notebook file Rules 3108-3149: 4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2)
PDF file Notebook file Rules 3150-3190: 4.1.5 trig^m (a cos+b sin)^n
PDF file Notebook file Rules 3191-3247: 4.1.6 (a+b cos+c sin)^n
PDF file Notebook file Rules 3248-3320: 4.1.7 (d trig)^m (a+b (c sin)^n)^p
PDF file Notebook file Rules 3321-3324: 4.1.8 trig^m (a+b cos^p+c sin^q)^n
PDF file Notebook file Rules 3325-3376: 4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p
PDF file Notebook file Rules 3377-3409: 4.1.10 (c+d x)^m (a+b sin)^n
PDF file Notebook file Rules 3410-3431: 4.1.11 (e x)^m (a+b x^n)^p sin
PDF file Notebook file Rules 3432-3525: 4.1.12 (e x)^m (a+b sin(c+d x^n))^p
PDF file Notebook file Rules 3526-3553: 4.1.13 (d+e x)^m sin(a+b x+c x^2)^n

4.3 Tangent and cotangent

PDF file Notebook file Rules 3554-3566: 4.3.1.1 (a+b tan)^n
PDF file Notebook file Rules 3567-3596: 4.3.1.2 (d sec)^m (a+b tan)^n
PDF file Notebook file Rules 3597-3602: 4.3.1.3 (d sin)^m (a+b tan)^n
PDF file Notebook file Rules 3603-3660: 4.3.2.1 (a+b tan)^m (c+d tan)^n
PDF file Notebook file Rules 3661-3668: 4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n
PDF file Notebook file Rules 3669-3697: 4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan)
PDF file Notebook file Rules 3698-3712: 4.3.4.1 (a+b tan)^m (A+B tan+C tan^2)
PDF file Notebook file Rules 3713-3737: 4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2)
PDF file Notebook file Rules 3738-3762: 4.3.7 (d trig)^m (a+b (c tan)^n)^p
PDF file Notebook file Rules 3763-3796: 4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p
PDF file Notebook file Rules 3797-3823: 4.3.10 (c+d x)^m (a+b tan)^n
PDF file Notebook file Rules 3824-3843: 4.3.11 (e x)^m (a+b tan(c+d x^n))^p
PDF file Notebook file Rules 3844-3851: 4.3.12 (d+e x)^m tan(a+b x+c x^2)^n

4.5 Secant and cosecant

PDF file Notebook file Rules 3852-3871: 4.5.1.1 (a+b sec)^n
PDF file Notebook file Rules 3872-3956: 4.5.1.2 (d sec)^n (a+b sec)^m
PDF file Notebook file Rules 3957-3963: 4.5.1.3 (d sin)^n (a+b sec)^m
PDF file Notebook file Rules 3964-3987: 4.5.1.4 (d tan)^n (a+b sec)^m
PDF file Notebook file Rules 3988-4034: 4.5.2.1 (a+b sec)^m (c+d sec)^n
PDF file Notebook file Rules 4035-4080: 4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n
PDF file Notebook file Rules 4081-4125: 4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec)
PDF file Notebook file Rules 4126-4156: 4.5.4.1 (a+b sec)^m (A+B sec+C sec^2)
PDF file Notebook file Rules 4157-4204: 4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2)
PDF file Notebook file Rules 4205-4236: 4.5.7 (d trig)^m (a+b (c sec)^n)^p
PDF file Notebook file Rules 4237-4264: 4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p
PDF file Notebook file Rules 4265-4280: 4.5.10 (c+d x)^m (a+b sec)^n
PDF file Notebook file Rules 4281-4298: 4.5.11 (e x)^m (a+b sec(c+d x^n))^p

4.7 Miscellaneous trig functions

PDF file Notebook file Rules 4299-4323: 4.7.1 Sine normalization rules
PDF file Notebook file Rules 4324-4342: 4.7.2 Tangent normalization rules
PDF file Notebook file Rules 4343-4366: 4.7.3 Secant normalization rules
PDF file Notebook file Rules 4367-4416: 4.7.4 (c trig)^m (d trig)^n
PDF file Notebook file Rules 4417-4488: 4.7.5 Inert trig functions
PDF file Notebook file Rules 4489-4516: 4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p
PDF file Notebook file Rules 4517-4559: 4.7.7 F^(c (a+b x)) trig(d+e x)^n
PDF file Notebook file Rules 4560-4607: 4.7.8 u trig(a+b log(c x^n))^p
PDF file Notebook file Rules 4608-4711: 4.7.9 Active trig functions

5 Inverse Trig Function Rules

5.1 Inverse sine and cosine

PDF file Notebook file Rules 4712-4717: 5.1.1 (a+b arcsin(c x))^n
PDF file Notebook file Rules 4718-4731: 5.1.2 (d x)^m (a+b arcsin(c x))^n
PDF file Notebook file Rules 4732-4761: 5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n
PDF file Notebook file Rules 4762-4821: 5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n
PDF file Notebook file Rules 4822-4883: 5.1.5 u (a+b arcsin(c x))^n
PDF file Notebook file Rules 4884-4926: 5.1.6 Miscellaneous inverse sine

5.3 Inverse tangent and cotangent

PDF file Notebook file Rules 4927-4936: 5.3.1 (a+b arctan(c x^n))^p
PDF file Notebook file Rules 4937-4960: 5.3.2 (d x)^m (a+b arctan(c x^n))^p
PDF file Notebook file Rules 4961-4982: 5.3.3 (d+e x)^m (a+b arctan(c x^n))^p
PDF file Notebook file Rules 4983-5143: 5.3.4 u (a+b arctan(c x))^p
PDF file Notebook file Rules 5144-5163: 5.3.5 u (a+b arctan(c+d x))^p
PDF file Notebook file Rules 5164-5239: 5.3.6 Exponentials of inverse tangent
PDF file Notebook file Rules 5240-5318: 5.3.7 Miscellaneous inverse tangent

5.5 Inverse secant and cosecant

PDF file Notebook file Rules 5319-5354: 5.5.1 u (a+b arcsec(c x))^n
PDF file Notebook file Rules 5355-5378: 5.5.2 Miscellaneous inverse secant

6 Hyperbolic Function Rules

6.1 Hyperbolic sine and cosine

PDF file Notebook file Rules 5379-5380: 6.1.10 (c+d x)^m (a+b sinh)^n
PDF file Notebook file Rules 5381-5402: 6.1.11 (e x)^m (a+b x^n)^p sinh
PDF file Notebook file Rules 5403-5478: 6.1.12 (e x)^m (a+b sinh(c+d x^n))^p
PDF file Notebook file Rules 5479-5502: 6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n

6.3 Hyperbolic tangent and cotangent

PDF file Notebook file Rules 5503-5504: 6.3.10 (c+d x)^m (a+b tanh)^n
PDF file Notebook file Rules 5505-5524: 6.3.11 (e x)^m (a+b tanh(c+d x^n))^p
PDF file Notebook file Rules 5525-5530: 6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n

6.5 Hyperbolic secant and cosecant

PDF file Notebook file Rules 5531-5532: 6.5.10 (c+d x)^m (a+b sech)^n
PDF file Notebook file Rules 5533-5550: 6.5.11 (e x)^m (a+b sech(c+d x^n))^p

6.7 Miscellaneous hyperbolic functions

PDF file Notebook file Rules 5551-5578: 6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p
PDF file Notebook file Rules 5579-5619: 6.7.7 F^(c (a+b x)) hyper(d+e x)^n
PDF file Notebook file Rules 5620-5669: 6.7.8 u hyper(a+b log(c x^n))^p
PDF file Notebook file Rules 5670-5765: 6.7.9 Active hyperbolic functions

7 Inverse Hyperbolic Function Rules

7.1 Inverse hyperbolic sine

PDF file Notebook file Rules 5766-5768: 7.1.1 (a+b arcsinh(c x))^n
PDF file Notebook file Rules 5769-5775: 7.1.2 (d x)^m (a+b arcsinh(c x))^n
PDF file Notebook file Rules 5776-5790: 7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n
PDF file Notebook file Rules 5791-5820: 7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n
PDF file Notebook file Rules 5821-5851: 7.1.5 u (a+b arcsinh(c x))^n
PDF file Notebook file Rules 5852-5872: 7.1.6 Miscellaneous inverse hyperbolic sine

7.2 Inverse hyperbolic cosine

PDF file Notebook file Rules 5873-5875: 7.2.1 (a+b arccosh(c x))^n
PDF file Notebook file Rules 5876-5882: 7.2.2 (d x)^m (a+b arccosh(c x))^n
PDF file Notebook file Rules 5883-5905: 7.2.3 (d+e x^2)^p (a+b arccosh(c x))^n
PDF file Notebook file Rules 5906-5955: 7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n
PDF file Notebook file Rules 5956-5988: 7.2.5 u (a+b arccosh(c x))^n
PDF file Notebook file Rules 5989-6014: 7.2.6 Miscellaneous inverse hyperbolic cosine

7.3 Inverse hyperbolic tangent and cotangent

PDF file Notebook file Rules 6015-6024: 7.3.1 (a+b arctanh(c x^n))^p
PDF file Notebook file Rules 6025-6048: 7.3.2 (d x)^m (a+b arctanh(c x^n))^p
PDF file Notebook file Rules 6049-6070: 7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p
PDF file Notebook file Rules 6071-6231: 7.3.4 u (a+b arctanh(c x))^p
PDF file Notebook file Rules 6232-6251: 7.3.5 u (a+b arctanh(c+d x))^p
PDF file Notebook file Rules 6252-6333: 7.3.6 Exponentials of inverse hyperbolic tangent
PDF file Notebook file Rules 6334-6405: 7.3.7 Miscellaneous inverse hyperbolic tangent

7.5 Inverse hyperbolic secant and cosecant

PDF file Notebook file Rules 6406-6441: 7.5.1 u (a+b arcsech(c x))^n
PDF file Notebook file Rules 6442-6477: 7.5.2 Miscellaneous inverse hyperbolic secant

8 Special Function Rules

PDF file Notebook file Rules 6478-6546: 8.1 Error functions
PDF file Notebook file Rules 6547-6601: 8.2 Fresnel integral functions
PDF file Notebook file Rules 6602-6627: 8.3 Exponential integral functions
PDF file Notebook file Rules 6628-6656: 8.4 Trig integral functions
PDF file Notebook file Rules 6657-6685: 8.5 Hyperbolic integral functions
PDF file Notebook file Rules 6686-6709: 8.6 Gamma functions
PDF file Notebook file Rules 6710-6714: 8.7 Zeta function
PDF file Notebook file Rules 6715-6740: 8.8 Polylogarithm function
PDF file Notebook file Rules 6741-6784: 8.9 Product logarithm function

9 Miscellaneous Rules

PDF file Notebook file Rules 0001-0028: 9.1 Integrand simplification rules
PDF file Notebook file Rules 6785-6805: 9.2 Derivative integration rules
PDF file Notebook file Rules 2188-2206: 9.3 Piecewise linear functions
PDF file Notebook file Rules 6806-6872: 9.4 Miscellaneous integration rules

Rubi (Rule-based Integrator)

A symbolic integrator based on an extensive system of integration rules