Ten Hard Integrals. Nasser M. Abbasi. page compiled on November 8, 2015 at 8:25pm
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1 Ten Hard Integrals Nasser M. Abbasi March 5, 2014 page compiled on November 8, 2015 at 8:25pm 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 (Maxima ) on windows 7 7. Axiom on windows 7 (May 2012) welcome screen image 8. sympy on linux (python full installation. sympy rc1-3) starting image 9. reduce reduce-windows 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 Mathematica notebook of the first 20 integrals Mathematica PDF of the first 20 integrals Mathematica HTML of the first 20 integrals 2. Rubi downloads Rubi notebook for the first 20 integrals Rubi PDF for the first 20 integrals Rubi HTML for the first 20 integrals 3. Maple downloads Maple worksheet in wm format HTML worksheet in HTML 1
2 PDF worksheet in PDF Maple Problem 10 trace entered as int(x^3*exp(1)^arcsin(x)/sqrt(1-x^2),x); HTML PDF mw Maple worksheet Maple Problem 10 trace entered as int(x^3*exp(arcsin(x))/sqrt(1-x^2),x); HTML PDF mw Maple worksheet 4. maxima.wxm Maxima notebook. 5. copy of Kevin Charlwood s 2008 paper in PDF 6. Albert Rich pdf file showing 50 integrals and the best antiderivatives expected 7. arich/charlwoodproblems.m The above is m format xcas web page 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 score Mathematica 9.01 ( ) ( ) ( ) ( ) ( ) 100% Rubi 4.1 ( ) 90% Maple 18 ( ) 90% Axiom (May 2012) 80% FriCAS % Sage % Maxima % xcas % Sympy % Reduce % mupad 2013a 20% 2
3 1 arcsin(x) ln(x) dx optimal 2 1 x 2 + ( 1 x 2 log(x) + tanh 1 ) 1 x 2 x(1 log(x)) sin 1 (x) M 9.01 Rubi 4.1 Maple 18 Sage 5.4 3
4 Fricas 1.2 sympy Axiom 5/12 Maxima reduce 2008 mupad 2013a HTML of trace PDF of trace evalin(symengine,'int(asin(x)*log(x),x)') int(asin(x)*log(x), x) 4
5 xcas 5
6 2 x arcsin(x) 1 x 2 dx optimal x 1 x 2 sin 1 (x) M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy
7 Axiom 5/12 Maxima reduce 2008 mupad 2013a evalin(symengine,'int(x*asin(x)/sqrt(1-x^2),x)') int((x*asin(x))/(1 - x^2)^(1/2), x) xcas 7
8 3 ( x ) arcsin + 1 x dx optimal ( ) x+3 x+1 x x+1 x 4 2 ( x + 3 ) 8 sin 1 ( x x + 1 ) M 9.01 Rubi 4.1 Maple 18 8
9 Sage 5.4 Fricas 1.2 sympy Axiom 5/12 Maxima reduce 2008 mupad 2013a evalin(symengine,'int(asin(sqrt(x+1)-sqrt(x)),x)') -int(asin(x^(1/2) - (x + 1)^(1/2)), x) 9
10 xcas 10
11 4 ln (1 + x ) 1 + x 2 dx ( x ) optimal x log 2 + 1x ( ) ( ( x )) tan x 2 ( 5 1 ) ( tanh 1 ( x )) x 2x M
12 Rubi
13 Maple 18 Sage
14 Fricas
15 sympy
16 Axiom 5/12 16
17 Maxima reduce 2008 mupad 2013a evalin(symengine,'int(log(1+x*sqrt(1+x^2)),x)') result in mupad_4.txt xcas 17
18 5 cos 2 (x) cos4 (x) + cos 2 (x) + 1 dx optimal ( x tan 1 sin(x) cos(x) ( cos 2 (x)+1 ) cos 4 (x)+cos 2 (x)+1 cos 2 (x)+1 ) M 9.01 Rubi
19 Maple 18 Sage 5.4 Fricas 1.2 sympy (waited one hr) 19
20 Axiom 5/12 Maxima reduce 2008 mupad 2013a evalin(symengine,'int(cos(x)^2/sqrt(cos(x)^4+cos(x)^2+1),x)') int(cos(x)^2/(cos(x)^4 + cos(x)^2 + 1)^(1/2), x) xcas 20
21 6 tan(x) 1 + tan 4 (x) dx optimal 1 2 tan 4 (x) + 1 ( ) tanh 1 1 tan 2 (x) 2 tan 4 (x) sinh 1 ( tan 2 (x) ) M 9.01 click to view Rubi 4.1 Maple 18 Sage
22 Fricas 1.2 sympy
23 Axiom 5/12 Maxima
24 reduce 2008 mupad 2013a evalin(symengine,'int(tan(x)*sqrt(1+tan(x)^4),x)') int(tan(x)*(tan(x)^4 + 1)^(1/2), x) xcas 24
25 7 tan(x) sec3 (x) + 1 dx ( ) optimal 2 3 tanh 1 sec 3 (x) + 1 M 9.01 Rubi 4.1 Maple 18 Sage
26 Fricas 1.2 sympy Axiom 5/12 Maxima
27 reduce 2008 mupad 2013a evalin(symengine,'int(tan(x)/sqrt(sec(x)^3+1),x)') int(tan(x)/(1/cos(x)^3 + 1)^(1/2), x) xcas 27
28 8 tan 2 (x) + 2 tan(x) + 2 dx optimal 1 2 ( ( ) tan tan(x) 5 1 ( ) 1 ) 2 tan(x)(tan(x)+2) tanh 1 sinh 1 (tan(x) + 1) ( 5 1 tan(x) tan(x)(tan(x)+2) M 9.01 click to view Rubi 4.1 Maple 18 image Sage 5.4 Fricas 1.2 sympy Axiom 5/12 HTML Maxima
29 reduce 2008 mupad 2013a evalin(symengine,'int(sqrt(tan(x)^2+2*tan(x)+2),x)') int((2*tan(x) + tan(x)^2 + 2)^(1/2), x) xcas 29
30 9 ( ) sin(x) arctan sec(x) 1 dx optimal 1 2 cos(x) sec(x) tan 1 ( sec(x) 1 ) cos(x) tan 1 ( sec(x) 1 ) M 9.01 click to view Rubi 4.1 Maple 18 Sage
31 Fricas 1.2 update: per post on sci.math.symbolic on June 20, BTW: Current developement FriCAS can also do #9: (3) -> integrate(sin(x)*atan(sqrt(sec(x) - 1)), x) / (3) cos(x) + 1 cos(x) \ cos(x) - 2cos(x)atan(\ sec(x) - 1 ) + atan( ) cos(x) cos(x) + 1 cos(x) \ cos(x) Type: Union(Expression(Integer) sympy Axiom 5/12 Maxima
32 reduce 2008 evalin(symengine,'int(sin(x)*atan(sqrt(sec(x)-1)),x)') mupad 2013a pretty(ans) / 1/2 3 asin(cos(x) ) 3 (1 cos(x) /2 / / 1 \1/2 \ \ 2 cos(x) - atan cos(x) \ \ cos(x) / / / \ cos(x) xcas atan ( ( cos(x)) 2 +cos(x)sign(cos(x)) cos(x) 1 1 (sign(cos(x))) abs(sign(cos(x))) atan cos(x) 2 ) cos (x) + 2( sign (cos (x)) 2 asin (2 cos (x) + 1) + 2 (sign (cos (x)))3 1 (sign(cos(x))) 2 (2 ( cos(x)) 2 +cos(x) 1) (sign(cos(x))) cos(x) 1 abs(sign(cos(x))) 2 (sign(cos(x))) 2 1 ( cos(x)) 2 +cos(x) cos(x) 1 32
33 10 x 3 e arcsin(x) 1 x 2 dx ( 1 optimal 10 x x 2 x 2 3 ) 1 x 2 + 3x e sin 1 (x) M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas
34 sympy Axiom 5/12 Maxima reduce 2008 mupad 2013a evalin(symengine,'int(x^3*exp(asin(x))/sqrt(1-x^2),x)') int((x^3*exp(asin(x)))/(1 - x^2)^(1/2), x) xcas 34
35 11 x log(1 + x 2 ) log(x x 2 ) 1 + x 2 dx optimal M 9.01 Rubi 4.1 Maple 18 35
36 12 ( ) tan 1 1 x2 + x dx optimal M
37 Rubi 4.1 Maple 18 37
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