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Ten Hard Integrals

Nasser M. Abbasi

March 5, 2014 page compiled on November 8, 2015 at 8:26pm

The first 10 integrals from Kevin Charlwood's 2008 article ”Integration on Computer Algebra Systems” are solved using different CAS systems.

The original post on this topic is sci.math.symbolic by Martin

These are the CAS systems used

1.
Maple 18 on windows 7 (64 bit)
2.
Mathematica 9.01 on windows 7
3.
Rubi 4.1 on Mathematica 9.01 on windows 7
4.
Sage 5.4 using the Sage web server notebook interface
5.
Fricas 1.2 on Linux using sbcl lisp
6.
wxMaxima 12.04.0 (Maxima 5.28.02) on windows 7
7.
Axiom on windows 7 (May 2012) welcome screen image
8.
sympy on linux (python 2.7.3 full installation. sympy 0.7.1.rc1-3) starting image
9.
reduce reduce-windows64-20110414 help screen message with the algint package loaded.
10.
mupad engine in Matlab 2013a symbolic toolbox
11.
xcas 2013 January, version 1.0 on windows 7

Optimal answer to each intergal taken from Rich's referenced PDF below.

Downloads, references and links

1.
Mathematica downloads
2.
Rubi downloads
3.
Maple downloads

Maple Problem 10 trace entered as int(x^3*exp(1)^arcsin(x)/sqrt(1-x^2),x);

Maple Problem 10 trace entered as int(x^3*exp(arcsin(x))/sqrt(1-x^2),x);

4.
maxima.wxm Maxima notebook.
5.
copy of Kevin Charlwood's 2008 paper in PDF
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http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf Albert Rich pdf file showing 50 integrals and the best antiderivatives expected
7.
http://www.apmaths.uwo.ca/ arich/CharlwoodProblems.m The above is m format.
8.
http://www.math.utah.edu/faq/reduce/
9.
http://www.reduce-algebra.com/packages.htm
10.
http://reduce-algebra.sourceforge.net/
11.
http://www.reduce-algebra.com/docs/reduce.pdf
12.
xCAS web page
13.
http://www.apmaths.uwo.ca/~arich/ Rubi Mathematica package home

The following is summary of results for each integral. Result with a () around it means the antiderivative contains nonelementary functions.













system 1 2 3 4 5 6 7 8 9 10 score
























Mathematica 9.01 () () () () () 100%












Rubi 4.1 () 90%












Maple 18 () 90%












Axiom (May 2012) 80%












FriCAS 1.2 80%












Sage 5.4 40%












Maxima 5.28.02 40%












xcas 1.0 40%












Sympy 0.7.2 20%












Reduce 2008 20%












mupad 2013a 20%












1 ∫ arcsin(x )ln(x)dx
2 ∫
   xa√rcsin(x)dx
      1 − x2
3 ∫
         (√ ------  √ -)
   arcsin    x + 1 −   x  dx
4 ∫    (     √ ------)
   ln  1 + x  1 + x2  dx
5 ∫             2
   ∘-------cos-(x-)--------dx
     cos4(x) + cos2(x ) + 1
6 ∫        ∘ ------------
   tan (x)  1 + tan4(x)dx
7 ∫
  ∘---tan(x)----dx
     sec3(x ) + 1
8 ∫ ∘  ----------------------
     tan2 (x) + 2 tan(x) + 2dx
9 ∫              (∘  ---------)
  sin(x)arctan     sec(x ) − 1 dx
10 ∫
   x3earcsin(x)
   √1--−-x2-dx
11 ∫            2         √ -----2-
   xlog(1-+-x√)log(x-+---1-+-x-)dx
               1 + x2
12 ∫
      −1( √------2    )
  tan      1 − x  + x  dx