Here's this in Sage. See the end of this post for a solution you
might like better than the ramblings in between :)
----------------------------------------------------------------------
| Sage Version 4.7.2, Release Date: 2011-10-29 |
| Type notebook() for the GUI, and license() for information. |
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sage: var('m r a R')
(m, r, a, R)
sage: assume(2*m-3==0)
^[[A^[ integrate(1/r^(2*m+2)*(2*(r/a)^2-(r/a)^4),r)
-2*r^(-2*m + 1)/((2*m - 1)*a^2) - log(r)/a^4
sage: forget()
sage: assume(2*m-1<0)
sage: integrate(1/r^(2*m+2)*(2*(r/a)^2-(r/a)^4),r)
-2*r^(-2*m + 1)/((2*m - 1)*a^2) + r^(-2*m + 3)/((2*m - 3)*a^4)
Just to make the point that this is something we can't really avoid,
using Maxima as the default integration engine.
Maxima 5.25.0 http://maxima.sourceforge.net
using Lisp SBCL 1.0.24
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) integrate(1/r^(2*m+2)*(2*(r/a)^2-(r/a)^4),r);
Is 2 m - 3 zero or nonzero?
zero
;
1 - 2 m
2 r log(r)
(%o1) ------------ - ------
2 4
a (1 - 2 m) a
(%i2) integrate(1/r^(2*m+2)*(2*(r/a)^2-(r/a)^4),r);
Is 2 m - 3 zero or nonzero?
nonzero;
Is 2 m - 1 zero or nonzero?
nonzero;
1 - 2 m 3 - 2 m
2 r r
(%o2) ------------ - ------------
2 4
a (1 - 2 m) a (3 - 2 m)
"Bottom line, sage currently has to be coddled excessively via
assumptions into giving any result at all, verses system which give
results which may happen to be incorrect for certain values. "
But that sounds dangerous to me, at least for this particular
example. How is it different from this example?
sage: var('n')
n
sage: integrate(x^n,x)
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before integral evaluation
*may* help (example of legal syntax is 'assume(n+1>0)', see `assume?`
for more details)
Is n+1 zero or nonzero?
If anything, we've probably been too cavalier with our use of Maxima's
fine-grained simplification and other routines at times; it seems
dangerous to go the other way.
At the same time, we have
sage: integrate(x^n,x,algorithm='sympy')
x^(n + 1)/(n + 1)
which will at least sometimes do what you are interested in. See
the PS for the fact that it doesn't always do so.
- kcrisman
PS to devs:
However
sage: integrate(1/r^(2*m+2)*(2*(r/a)^2-(r/a)^4),r,algorithm='sympy')
---------------------------------------------------------------------------
AttributeError Traceback (most recent call
last)
v, a, b)
37 else:
38 result = sympy.integrate(ex, (v, a._sympy_(),
b._sympy_()))
---> 39 return result._sage_()
40
41 def mma_free_integrator(expression, v, a=None, b=None):
AttributeError: 'Integral' object has no attribute '_sage_'
That looks like a bug in our sympy interface. Any takers?
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