This repository provides all Rubi integration rules in human readable form as PDF files. Rubi systematically applies precisely defined rules to efficiently integrate a large class of mathematical expressions. The rules are organized based on the type of the integrand and hierarchically arranged in the form of a decision tree. By answering true-or-false questions in the tree, it is easy for a human or computer to determine exactly which of the more than 6600 rules is the right one to apply to a given integrand.
The rules are displayed in mathematical notation followed by the equivalent Mathematica program code. The following types of comments precede many of the rules
- 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.
In the following, you will find the complete outline of the organized rules. The outline reflects the directory structure
under the PdfFiles directory. If you seek a human readable outline, please view the README.txt file.
- 1.1.1.1 (a+b x)^m
- 1.1.1.2 (a+b x)^m (c+d x)^n
- 1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p
- 1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q
- 1.1.1.5 P(x) (a+b x)^m (c+d x)^n
- 1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p
- 1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q
- 1.1.3.1 (a+b x^n)^p
- 1.1.3.2 (c x)^m (a+b x^n)^p
- 1.1.3.3 (a+b x^n)^p (c+d x^n)^q
- 1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q
- 1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r
- 1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r
- 1.1.3.7 P(x) (a+b x^n)^p
- 1.1.3.8 P(x) (c x)^m (a+b x^n)^p
- 1.1.4.1 (a x^j+b x^n)^p
- 1.1.4.2 (c x)^m (a x^j+b x^n)^p
- 1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q
- 1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p
- 1.2.1.1 (a+b x+c x^2)^p
- 1.2.1.2 (d+e x)^m (a+b x+c x^2)^p
- 1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p
- 1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p
- 1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q
- 1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q
- 1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2)
- 1.2.1.8 P(x) (a+b x+c x^2)^p
- 1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p
- 1.2.2.1 (a+b x^2+c x^4)^p
- 1.2.2.2 (d x)^m (a+b x^2+c x^4)^p
- 1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p
- 1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p
- 1.2.2.5 P(x) (a+b x^2+c x^4)^p
- 1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p
- 1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p
- 1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p
- 1.2.3.1 (a+b x^n+c x^(2 n))^p
- 1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p
- 1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p
- 1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p
- 1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p
- 1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p
- 1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p
- 1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p
- 1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p
- 1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p
- 1.3.1 P(x)^p
- 1.3.2 P(x) Q(x)^p
- 1.3.3 Miscellaneous algebraic functions
- 1.3.4 Normalizing algebraic functions
- 2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p
- 2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p
- 2.3 Miscellaneous exponentials
- 3.1 u (a+b log(c x^n))^p
- 3.2 u (a+b log(c (d+e x)^n))^p
- 3.3 u (a+b log(c (d+e x^m)^n))^p
- 3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s
- 3.5 Miscellaneous logarithms
- 4.1.0.1 (a sin)^m (b trg)^n
- 4.1.0.2 (a trg)^m (b tan)^n
- 4.1.0.3 (a csc)^m (b sec)^n
- 4.1.10 (c+d x)^m (a+b sin)^n
- 4.1.1.1 (a+b sin)^n
- 4.1.11 (e x)^m (a+b x^n)^p sin
- 4.1.12 (e x)^m (a+b sin(c+d x^n))^p
- 4.1.1.2 (g cos)^p (a+b sin)^m
- 4.1.13 (d+e x)^m sin(a+b x+c x^2)^n
- 4.1.1.3 (g tan)^p (a+b sin)^m
- 4.1.2.1 (a+b sin)^m (c+d sin)^n
- 4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n
- 4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n
- 4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin)
- 4.1.4.1 (a+b sin)^m (A+B sin+C sin^2)
- 4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2)
- 4.1.5 trig^m (a cos+b sin)^n
- 4.1.6 (a+b cos+c sin)^n
- 4.1.7 (d trig)^m (a+b (c sin)^n)^p
- 4.1.8 trig^m (a+b cos^p+c sin^q)^n
- 4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p
- 4.3.10 (c+d x)^m (a+b tan)^n
- 4.3.1.1 (a+b tan)^n
- 4.3.11 (e x)^m (a+b tan(c+d x^n))^p
- 4.3.12 (d+e x)^m tan(a+b x+c x^2)^n
- 4.3.1.2 (d sec)^m (a+b tan)^n
- 4.3.1.3 (d sin)^m (a+b tan)^n
- 4.3.2.1 (a+b tan)^m (c+d tan)^n
- 4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n
- 4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan)
- 4.3.4.1 (a+b tan)^m (A+B tan+C tan^2)
- 4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2)
- 4.3.7 (d trig)^m (a+b (c tan)^n)^p
- 4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p
- 4.5.10 (c+d x)^m (a+b sec)^n
- 4.5.1.1 (a+b sec)^n
- 4.5.11 (e x)^m (a+b sec(c+d x^n))^p
- 4.5.1.2 (d sec)^n (a+b sec)^m
- 4.5.1.3 (d sin)^n (a+b sec)^m
- 4.5.1.4 (d tan)^n (a+b sec)^m
- 4.5.2.1 (a+b sec)^m (c+d sec)^n
- 4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n
- 4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec)
- 4.5.4.1 (a+b sec)^m (A+B sec+C sec^2)
- 4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2)
- 4.5.7 (d trig)^m (a+b (c sec)^n)^p
- 4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p
- 4.7.1 Sine normalization rules
- 4.7.2 Tangent normalization rules
- 4.7.3 Secant normalization rules
- 4.7.4 (c trig)^m (d trig)^n
- 4.7.5 Inert trig functions
- 4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p
- 4.7.7 F^(c (a+b x)) trig(d+e x)^n
- 4.7.8 u trig(a+b log(c x^n))^p
- 4.7.9 Active trig functions
- 5.1.1 (a+b arcsin(c x))^n
- 5.1.2 (d x)^m (a+b arcsin(c x))^n
- 5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n
- 5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n
- 5.1.5 u (a+b arcsin(c x))^n
- 5.1.6 Miscellaneous inverse sine
- 5.3.1 u (a+b arctan(c x^n))^p
- 5.3.2 u (a+b arctan(c+d x))^p
- 5.3.3 Exponentials of inverse tangent
- 5.3.4 Miscellaneous inverse tangent
- 6.1.10 (c+d x)^m (a+b sinh)^n
- 6.1.11 (e x)^m (a+b x^n)^p sinh
- 6.1.12 (e x)^m (a+b sinh(c+d x^n))^p
- 6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n
- 6.3.10 (c+d x)^m (a+b tanh)^n
- 6.3.11 (e x)^m (a+b tanh(c+d x^n))^p
- 6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n
- 6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p
- 6.7.7 F^(c (a+b x)) hyper(d+e x)^n
- 6.7.8 u hyper(a+b log(c x^n))^p
- 6.7.9 Active hyperbolic functions
- 7.1.1 (a+b arcsinh(c x))^n
- 7.1.2 (d x)^m (a+b arcsinh(c x))^n
- 7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n
- 7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n
- 7.1.5 u (a+b arcsinh(c x))^n
- 7.1.6 Miscellaneous inverse hyperbolic sine
- 7.3.1 u (a+b arctanh(c x^n))^p
- 7.3.2 u (a+b arctanh(c+d x))^p
- 7.3.3 Exponentials of inverse hyperbolic tangent
- 7.3.4 Miscellaneous inverse hyperbolic tangent