# Rubi Integration Test Suite

## Expressed In Axiom Syntax

This page provides links to text files containing the Rubi Integration Test Suite expressed in Axiom syntax. Each problem is stored as a list of the following items:

1. the integrand;
2. the integration variable;
3. the number of rule applications required to integrate the integrand; and
4. the optimal antiderivative.

The optimal antiderivatives in the test suite are just the simplest ones found so far. If you find significantly simpler antiderivatives and want to contribute them to the test suite, please email them to Albert Rich.

To download the test suite files, click on one of the following links:

Integration problems from independent test suites:

Integration problems involving rational and algebraic functions:

Problems involving powers of linear binomials:

Problems involving powers of quadratic binomials:

Problems involving powers of general binomials:

Problems involving powers of improper binomials:

Problems involving powers of quadratic trinomials:

Problems involving powers of quartic trinomials:

Problems involving powers of general trinomials:

Problems involving powers of improper trinomials:

Miscellaneous rational and algebraic function problems:

• Download Problems involving miscellaneous rational functions
• Download Problems involving miscellaneous algebraic functions
• Download Problems requiring integration by substitution
• Download Problems requiring piecewise linear extraction

Integration problems involving exponentials:

• Download (c+d x)m (F^(g (e+f x)))n (a+b (F^(g (e+f x)))^n)p

Integration problems involving logarithms:

• Download Problems involving (a+b log(c xn))p
• Download Problems involving (a+b log(c (d+e x)n))p
• Download Problems involving (a+b log(c (d+e xm)n))p
• Download Problems involving log(e (f (a+b x)p (c+d x)q)r)s

Integration problems involving trig functions:

Sine function problems:

• Download (a sin(c+d x))m (b trig(c+d x))n
• Download (g cos(e+f x))p (a+b sin(e+f x))m
• Download (g tan(e+f x))p (a+b sin(e+f x))m
• Download (a+b sin(e+f x))m (c+d sin(e+f x))n
• Download (g cos(e+f x))p (a+b sin(e+f x))m (c+d sin(e+f x))n
• Download (g sin(e+f x))p (a+b sin(e+f x))m (c+d sin(e+f x))n
• Download (a+b sin(e+f x))m (c+d sin(e+f x))n (A+B sin(e+f x))
• Download (a+b sin(e+f x))m (A+B sin(e+f x)+C sin(e+f x)2)
• Download (a+b sin(e+f x))m (c+d sin(e+f x))n (A+B sin(e+f x)+C sin(e+f x)2)
• Download (d trig(e+f x))m (a+b (c sin(e+f x))n)p
• Download (a+b sin(e+f x))m (c+d trig(e+f x))n
• Download trig(d+e x)m (a+b sin(d+e x)n+c sin(d+e x)2 n)p
• Download (c+d x)m (a+b sin(e+f x))n
• Download (e x)m (a+b xn)p sin(c+d x)
• Download (e x)m (a+b sin(c+d xn))p
• Download (d+e x)m sin(a+b x+c x2)n

Cosine function problems:

• Download (a cos(c+d x))m (b trig(c+d x))n
• Download (g sin(e+f x))p (a+b cos(e+f x))m
• Download (g tan(e+f x))p (a+b cos(e+f x))m
• Download (a+b cos(e+f x))m (c+d cos(e+f x))n
• Download (g sin(e+f x))p (a+b cos(e+f x))m (c+d cos(e+f x))n
• Download (g cos(e+f x))p (a+b cos(e+f x))m (c+d cos(e+f x))n
• Download (a+b cos(e+f x))m (c+d cos(e+f x))n (A+B cos(e+f x))
• Download (a+b cos(e+f x))m (A+B cos(e+f x)+C cos(e+f x)2)
• Download (a+b cos(e+f x))m (c+d cos(e+f x))n (A+B cos(e+f x)+C cos(e+f x)2)
• Download (d trig(e+f x))m (a+b (c cos(e+f x))n)p
• Download (a+b cos(c+d x))m (c+d trig(c+d x))n
• Download trig(d+e x)m (a+b cos(d+e x)n+c cos(d+e x)2 n)p
• Download (c+d x)m (a+b cos(e+f x))n
• Download (e x)m (a+b cos(c+d xn))p
• Download (d+e x)m cos(a+b x+c x2)n

Tangent function problems:

• Download (a trig(c+d x))m (b tan(c+d x))n
• Download (d sec(c+d x))m (a+b tan(c+d x))n
• Download (d sin(c+d x))m (a+b tan(c+d x))n
• Download (a+b tan(e+f x))m (c+d tan(e+f x))n
• Download (a+b tan(e+f x))m (c+d tan(e+f x))n (A+B tan(e+f x))
• Download (a+b tan(e+f x))m (c+d tan(e+f x))n (A+B tan(e+f x)+C tan(e+f x)2)
• Download (d trig(e+f x))m (a+b (c tan(e+f x))n)p
• Download trig(d+e x)m (a+b tan(d+e x)n+c tan(d+e x)2 n)p
• Download (c+d x)m (a+b tan(e+f x))n
• Download (e x)m (a+b tan(c+d xn))p

Cotangent function problems:

• Download (a trig(c+d x))m (b cot(c+d x))n
• Download (d csc(c+d x))m (a+b cot(c+d x))n
• Download (d cos(c+d x))m (a+b cot(c+d x))n
• Download (a+b cot(e+f x))m (c+d cot(e+f x))n
• Download (d trig(e+f x))m (a+b (c cot(e+f x))n)p
• Download trig(d+e x)m (a+b cot(d+e x)n+c cot(d+e x)2 n)p
• Download (c+d x)m (a+b cot(e+f x))n

Secant function problems:

• Download (a sec(c+d x))m (b trig(c+d x))n
• Download (d sec(e+f x))n (a+b sec(e+f x))m
• Download (d sin(c+d x))n (a+b sec(c+d x))m
• Download (d tan(c+d x))n (a+b sec(c+d x))m
• Download (a+b sec(e+f x))m (c+d sec(e+f x))n
• Download (g sec(e+f x))p (a+b sec(e+f x))m (c+d sec(e+f x))n
• Download (a+b sec(e+f x))m (d sec(e+f x))n (A+B sec(e+f x))
• Download (a+b sec(e+f x))m (A+B sec(e+f x)+C sec(e+f x)2)
• Download (a+b sec(e+f x))m (d sec(e+f x))n (A+B sec(e+f x)+C sec(e+f x)2)
• Download (d trig(e+f x))m (a+b (c sec(e+f x))n)p
• Download (e x)m (a+b sec(c+d xn))p

Cosecant function problems:

• Download (a csc(c+d x))m (b trig(c+d x))n
• Download (d csc(e+f x))n (a+b csc(e+f x))m
• Download (d cos(c+d x))n (a+b csc(c+d x))m
• Download (d cot(c+d x))n (a+b csc(c+d x))m
• Download (a+b csc(e+f x))m (d csc(e+f x))n (A+B csc(e+f x))
• Download (a+b csc(e+f x))m (d csc(e+f x))n (A+B csc(e+f x)+C csc(e+f x)2)
• Download (d trig(e+f x))m (a+b (c csc(e+f x))n)p
• Download (e x)m (a+b csc(c+d xn))p

Miscellaneous trig function problems:

• Download (c trig(a+b x))m (c trig(a+b x))n
• Download trig(c+d x)m (a trig(c+d x)+b trig(c+d x))n
• Download (c+d x)m trig(a+b x)n trig(a+b x)p
• Download f^(a+b x+c x2) trig(d+e x+f x2)n
• Download Miscellaneous problems involving trig functions

Integration problems involving inverse trig functions:

Inverse sine function problems:

• Download (d x)m (a+b arcsin(c x)n
• Download (f x)m (d-c2 d x2)p (a+b arcsin(c x)n
• Download (f x)m (d+e x2)p (a+b arcsin(c x)n

Inverse cosine function problems:

Inverse tangent function problems:

• Download Problems involving (a+b arctan(c xn)p
• Download Problems involving (a+b arctan(c+d x))p
• Download Problems involving E^(n arctan(a x))
• Download Problems involving exponentials of inverse tangents

Inverse cotangent function problems:

• Download Problems involving exponentials of inverse cotangents

Inverse secant function problems:

• Download Problems involving (a+b arcsec(c x))n

Inverse cosecant function problems:

• Download Problems involving (a+b arccsc(c x))n

Integration problems involving hyperbolic functions:

Hyperbolic sine function problems:

Hyperbolic cosine function problems:

Hyperbolic tangent function problems:

• Download (c+d x)m (a+b tanh(e+f x))n
• Download (d hyper(e+f x))m (a+b (c tanh(e+f x))n)p

Hyperbolic cotangent function problems:

• Download (c+d x)m (a+b coth(e+f x))n
• Download (d hyper(e+f x))m (a+b (c coth(e+f x))n)p

Hyperbolic secant function problems:

• Download (c+d x)m (a+b sech(e+f x))n
• Download (e x)m (a+b sech(c+d xn))p
• Download (d hyper(e+f x))m (a+b (c sech(e+f x))n)p

Hyperbolic cosecant function problems:

• Download (c+d x)m (a+b csch(e+f x))n
• Download (e x)m (a+b csch(c+d xn))p
• Download (d hyper(e+f x))m (a+b (c csch(e+f x))n)p

Miscellaneous hyperbolic function problems:

• Download Problems involving multiple hyperbolic functions

Integration problems involving inverse hyperbolic functions:

Inverse hyperbolic sine function problems:

• Download (d x)m (a+b arcsinh(c x)n
• Download (f x)m (d+c2 d x2)p (a+b arcsinh(c x)n
• Download (f x)m (d+e x2)p (a+b arcsinh(c x)n
• Download Problems involving inverse hyperbolic sines

Inverse hyperbolic cosine function problems:

• Download (d x)m (a+b arccosh(c x)n
• Download (f x)m (d-c2 d x2)p (a+b arccosh(c x)n
• Download (f x)m (d+e x2)p (a+b arccosh(c x)n
• Download Problems involving inverse hyperbolic cosines

Inverse hyperbolic tangent function problems:

• Download Problems involving (a+b arctanh(c xn)p
• Download Problems involving (a+b arctanh(c+d x))p
• Download Problems involving inverse hyperbolic tangents
• Download xm (c-a2 c x2)p E^(n arctanh(a x))
• Download Exponentials of inverse hyperbolic tangent functions

Inverse hyperbolic cotangent function problems:

• Download Problems involving inverse hyperbolic cotangents
• Download Exponentials of inverse hyperbolic cotangent functions

Inverse hyperbolic secant function problems:

• Download Problems involving (a+b arcsech(c x))n
• Download Problems involving inverse hyperbolic secants

Inverse hyperbolic cosecant function problems:

• Download Problems involving (a+b arccsch(c x))n
• Download Problems involving inverse hyperbolic cosecants

Integration problems involving special functions: