login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

Edit detail for Symbolic Integration revision 4 of 12

1 2 3 4 5 6 7 8 9 10 11 12
Editor: test1
Time: 2013/04/11 17:45:59 GMT+0
Note:

removed:
-
-From unknown Sat May 21 12:49:39 -0500 2005
-From: unknown
-Date: Sat, 21 May 2005 12:49:39 -0500
-Subject: 
-Message-ID: <20050521124939-0500@page.axiom-developer.org>
-
-\begin{axiom}
-int(x,x)
-\end{axiom}

removed:
-
-From unknown Sat May 21 12:51:59 -0500 2005
-From: unknown
-Date: Sat, 21 May 2005 12:51:59 -0500
-Subject: 
-Message-ID: <20050521125159-0500@page.axiom-developer.org>
-
-\begin{axiom}
-axiomintegrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
-\end{axiom}

removed:
-From unknown Sat Oct 22 19:04:53 -0500 2005
-From: unknown
-Date: Sat, 22 Oct 2005 19:04:53 -0500
-Subject: 
-Message-ID: <20051022190453-0500@page.axiom-developer.org>
-
-int(sqrt(x), x)

changed:
-int(sqrt(x), x);
\begin{axiom}
integrate(sqrt(x), x)
\end{axiom}

changed:
-integrate(a*x,x);
integrate(a*x,x)

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
integrate(x,x)

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

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

axiom
integrate(sqrt(x), x)

\label{eq10}{2 \  x \ {\sqrt{x}}}\over 3(10)
Type: Union(Expression(Integer),...)

axiom
integrate(a*x,x)

\label{eq11}{1 \over 2}\  a \ {{x}^{2}}(11)
Type: Polynomial(Fraction(Integer))