help links subscribe changes refresh edit

	FriCAS Problems
Polynomial Coefficients ←	↑	→ Unevaluated Expressions

Symbolic Integration

Edit detail for Symbolic Integration revision 4 of 15

	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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

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$

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$

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}$

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)),...)

... --unknown, Sat, 21 May 2005 12:50:47 -0500 reply

axiom

integrate(x,x)

$\label{eq7}{1 \over 2}\ {{x}^{2}}$

(7)

Type: Polynomial(Fraction(Integer))

... --unknown, Sat, 21 May 2005 12:52:20 -0500 reply

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

solvin --mcd, Fri, 26 Jun 2009 00:20:27 -0700 reply

axiom

integrate(sqrt(x), x)

$\label{eq10}{2 \ x \ {\sqrt{x}}}\over 3$

(10)

Type: Union(Expression(Integer),...)

teste --mcd, Fri, 26 Jun 2009 00:23:35 -0700 reply

axiom

integrate(a*x,x)

$\label{eq11}{1 \over 2}\ a \ {{x}^{2}}$

(11)

Type: Polynomial(Fraction(Integer))