## Ten Hard Integrals

March 5, 2014   Compiled on September 11, 2023 at 5:40pm

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
6. 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 $$\int \arcsin (x) \ln (x)\,dx$$

 optimal $$-2 \sqrt {1-x^2}+\sqrt {1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt {1-x^2}\right )-x (1-\log (x)) \sin ^{-1}(x)$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 reduce 2008 mupad 2013a   evalin(symengine,'int(asin(x)*log(x),x)')   int(asin(x)*log(x), x) xcas

### 2 $$\displaystyle \int \frac {x \arcsin (x)}{\sqrt {1-x^2}} \,dx$$

 optimal $$x-\sqrt {1-x^2} \sin ^{-1}(x)$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 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

### 3 $$\displaystyle \int \arcsin \left ( \sqrt {x+1} - \sqrt {x} \right ) \,dx$$

 optimal $$\frac {\left (\sqrt {x}+3 \sqrt {x+1}\right ) \sqrt {\sqrt {x} \sqrt {x+1}-x}}{4 \sqrt {2}}-\left (x+\frac {3}{8}\right ) \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 reduce 2008 mupad 2013a evalin(symengine,'int(asin(sqrt(x+1)-sqrt(x)),x)') -int(asin(x^(1/2) - (x + 1)^(1/2)), x) xcas

### 4 $$\displaystyle \int \ln \left ( 1+x\sqrt {1+x^2} \right ) \,dx$$

 optimal $$x \log \left (\sqrt {x^2+1} x+1\right )+\sqrt {2 \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\sqrt {5}-2} \left (\sqrt {x^2+1}+x\right )\right )-\sqrt {2 \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\sqrt {2+\sqrt {5}} \left (\sqrt {x^2+1}+x\right )\right )-2 x$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 reduce 2008 mupad 2013a evalin(symengine,'int(log(1+x*sqrt(1+x^2)),x)') result in mupad_4.txt xcas

### 5 $$\displaystyle \int \frac {\cos ^2(x)}{\sqrt {\cos ^4(x)+\cos ^2(x)+1}} \,dx$$

 optimal $$\frac {x}{3}+\frac {1}{3} \tan ^{-1}\left (\frac {\sin (x) \cos (x) \left (\cos ^2(x)+1\right )}{\sqrt {\cos ^4(x)+\cos ^2(x)+1} \cos ^2(x)+1}\right )$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 (waited one hr) Axiom 5/12 Maxima 5.28.02 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

### 6 $$\displaystyle \int \tan (x) \sqrt {1+\tan ^4(x)} \,dx$$

 optimal $$\frac {1}{2} \sqrt {\tan ^4(x)+1}-\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{\sqrt {2}}-\frac {1}{2} \sinh ^{-1}\left (\tan ^2(x)\right )$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 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

### 7 $$\displaystyle \int \frac {\tan (x)}{\sqrt {\sec ^3(x)+1}} \,dx$$

 optimal $$-\frac {2}{3} \tanh ^{-1}\left (\sqrt {\sec ^3(x)+1}\right )$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 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

### 8 $$\displaystyle \int \sqrt {\tan ^2(x)+2 \tan (x)+2} \,dx$$

 optimal $$\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {5}} \tan (x)-\sqrt {\sqrt {5}-1}}{\sqrt {2} \sqrt {\tan (x) (\tan (x)+2)+2}}\right )-\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {5}-1} \tan (x)+\sqrt {1+\sqrt {5}}}{\sqrt {2} \sqrt {\tan (x) (\tan (x)+2)+2}}\right )+\sinh ^{-1}(\tan (x)+1)$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 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

### 9 $$\displaystyle \int \sin (x) \arctan \left ( \sqrt {\sec (x)-1} \right ) \,dx$$

 optimal $$\frac {1}{2} \cos (x) \sqrt {\sec (x)-1}+\frac {1}{2} \tan ^{-1}\left (\sqrt {\sec (x)-1}\right )-\cos (x) \tan ^{-1}\left (\sqrt {\sec (x)-1}\right )$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 update: per post on sci.math.symbolic on June 20, 2013 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) - 1      +               +------------+               |- cos(x) + 1        cos(x) |------------              \|   cos(x)   /      2                                          Type: Union(Expression(Integer),...) sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 reduce 2008 mupad 2013a evalin(symengine,'int(sin(x)*atan(sqrt(sec(x)-1)),x)') pretty(ans)                                               /              1/2                  1/2 \                                               | 3 asin(cos(x)   )   3 (1 - cos(x))    |             1/2                                        cos(x) | ----------------- - ----------------- | (1 - cos(x))                                               |            3/2          2 cos(x)      |         / /   1        \1/2 \                 \    2 cos(x)                           /   - atan| | ------ - 1 |    | cos(x) - ----------------------------------------------------------------         \ \ cos(x)     /    /                                  /   1        \1/2                                                              3 | ------ - 1 |                                                                \ cos(x)     / xcas $$-\mathrm {atan}\left (\frac {\sqrt {-\left (-\cos \left (x\right )\right )^{2}+\cos \left (x\right )} \mathrm {sign}\left (\cos \left (x\right )\right ) \cos \left (x\right )}{\cos \left (x\right )^{2}}\right ) \cos \left (x\right )+2 (-\mathrm {sign}\left (\cos \left (x\right )\right )\cdot \frac {1}{\left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )^{2}-1}\cdot \frac {1}{2}\cdot \frac {1}{2} \mathrm {asin}\left (2\cdot -\cos \left (x\right )+1\right )+2 \left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )^{3}\cdot \frac {1}{\left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )^{2}-1}\cdot \frac {1}{4}\cdot \frac {1}{\mathrm {abs}\left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )} \mathrm {atan}\left (\frac {\frac {\left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )^{2} (2 \sqrt {-\left (-\cos \left (x\right )\right )^{2}+\cos \left (x\right )}-1)}{-2\cdot -\cos \left (x\right )-1}-\left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )^{2}+\frac {2 \sqrt {-\left (-\cos \left (x\right )\right )^{2}+\cos \left (x\right )}-1}{-2\cdot -\cos \left (x\right )-1}+1}{\mathrm {abs}\left (\mathrm {sign}\left (\cos \left (x\right )\right )\right )\cdot 2}\right ))$$

### 10 $$\displaystyle \int \frac {x^3 e^{\arcsin (x)} }{\sqrt {1-x^2}} \,dx$$

 optimal $$\frac {1}{10} \left (x^3-3 \sqrt {1-x^2} x^2-3 \sqrt {1-x^2}+3 x\right ) e^{\sin ^{-1}(x)}$$ M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 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

### 11 $$\displaystyle \int \frac { x \log (1+x^2) \log (x+ \sqrt {1+x^2}) }{ \sqrt {1+x^2}} \,dx$$

 optimal M 9.01 Rubi 4.1 Maple 18

### 12 $$\displaystyle \int \tan ^{-1}\left (\sqrt {1-x^2}+x\right ) \,dx$$

 optimal M 9.01 Rubi 4.1 Maple 18