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Edit detail for Symbolic Integration revision 3 of 12

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Editor: mcd
Time: 2009/06/26 00:23:36 GMT-7
Note: teste

added:

From mcd Fri Jun 26 00:23:35 -0700 2009
From: mcd
Date: Fri, 26 Jun 2009 00:23:35 -0700
Subject: teste
Message-ID: <20090626002335-0700@axiom-wiki.newsynthesis.org>

\begin{axiom}
integrate(a*x,x);
\end{axiom}

Errors in symbolic integration

AXIOM Examples

1)

axiom
integrate(sin(x)+sqrt(1-x^3),x)

\label{eq1}\int^{
\displaystyle
x}{{\left({\sqrt{-{\%A^3}+ 1}}+{\sin \left({\%A}\right)}\right)}\ {d \%A}}(1)
Type: Union(Expression(Integer),...)

int(sin(x)+sqrt(1-x^3),x);
reduce
\displaylines{\qdd
\frac{-5\cdot \cos 
      \(x
       

2)

axiom
integrate(sqrt(1-log(sin(x)^2)),x)
>> Error detected within library code: integrate: implementation incomplete (constant residues)

int(sqrt(1-log(sin(x)^2)),x);
reduce
\displaylines{\qdd
\int {\sqrt{
            -\ln 
            \(\sin 
              \(x
               

3)

axiom
integrate(sqrt(sin(1/x)),x)
>> Error detected within library code: integrate: implementation incomplete (constant residues)

That seems strange given the claims about the "completeness" of Axiom's integration algorithm! But to be fair, Maple also returns this integral unevaluated.

int(sqrt(sin(1/x)),x);
reduce
\displaylines{\qdd
\frac{2\cdot 
      \sqrt{\sin 
            \(\frac{1}{
                    x}
             

4)

axiom
integrate(sqrt(sin(x)),x)

\label{eq2}\int^{
\displaystyle
x}{{\sqrt{\sin \left({\%A}\right)}}\ {d \%A}}(2)
Type: Union(Expression(Integer),...)

int(sqrt(sin(x)),x);
reduce
\displaylines{\qdd
\int {\sqrt{\sin 
            \(x
             

For this Maple 9 gives the following result:


\label{eq3}
    -{\frac {\sqrt {1+\sin \left( x \right) }\sqrt {-2\,\sin \left( x
    \right) +2}\sqrt {-\sin \left( x \right) }}{\cos \left( x \right) \sqrt {\sin \left( x
    \right) }}} \times
    \
    \left( 2\,{\it EllipticE}
    \left( \sqrt {1+\sin \left( x \right) },1/2\,\sqrt {2} \right) -{\it 
    EllipticF} \left( \sqrt {1+\sin \left( x \right) },1/2\,\sqrt {2}
    \right)  \right)
    (3)

And Mathematica 4 gives:


\label{eq4}
    -2\,{\it EllipticE}(\frac{\frac{\pi }{2} - x}{2},2)
    (4)

symbolic integration
Tue, 22 Mar 2005 11:48:00 -0600 reply
axiom
integrate(exp(-x^2),x)

\label{eq5}{{\erf \left({x}\right)}\ {\sqrt{\pi}}}\over 2(5)
Type: Union(Expression(Integer),...)
Errorfunction
Wed, 23 Mar 2005 08:23:21 -0600 reply
axiom
integrate(exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)

\label{eq6}{2 \ {\sqrt{\pi}}}\over{{\sqrt{2}}\ {\sqrt{2 \  \pi}}}(6)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

axiom
int(x,x)
There are no exposed library operations named int but there are 5 unexposed operations with that name. Use HyperDoc Browse or issue )display op int to learn more about the available operations.
Cannot find a definition or applicable library operation named int with argument type(s) Variable(x) Variable(x)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

axiom
integrate(x,x)

\label{eq7}{1 \over 2}\ {x^2}(7)
Type: Polynomial(Fraction(Integer))

axiom
axiomintegrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
There are no library operations named axiomintegrate Use HyperDoc Browse or issue )what op axiomintegrate to learn if there is any operation containing " axiomintegrate " in its name.
Cannot find a definition or applicable library operation named axiomintegrate with argument type(s) Expression(Integer) SegmentBinding(OrderedCompletion(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

axiom
integrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)

\label{eq8}\mbox{\tt "failed"}(8)
Type: Union(fail: failed,...)

The answer should be:


\label{eq9}
15\,{\frac {\sqrt {\pi }}{\sqrt {\pi}}}
(9)

integrate(exp(x)/x^2) --unknown, Thu, 25 Aug 2005 05:57:53 -0500 reply
Axiom does not perform the integration (while it perform the integration of exp(x)/x ), but the integration can be given in terms of Ei(x)

integrate(exp(x)/x^2,x) --> Ei(x)-exp(x)/x

int(sqrt(x), x)

int(sqrt(x), x);

axiom
integrate(a*x,x);
Type: Polynomial(Fraction(Integer))