Thank you for the interesting example.
Unfortunately, it looks as though it's rather messy to simplify the form
'integrate' gives. Here is one way. (I've written out the steps
individually and labelled them to make it easy to see what's going on.)
qq: integrate(integrate(sqrt(x^2+y^2),x,0,1),y,0,1);
qq1: logarc(qq); /* transform asinh to log */
qq2: logcontract(qq1),logconcoeffp:lambda([ex],true);
qq3: expand(qq2);
qq4: logcontract(qq3),logconcoeffp:lambda([ex],true);
qq5: part(qq4,1,1); /* need to isolate this part to work on it */
load(sqdnst)$
qq6: sqrtdenest(qq5);
qq7: sqrtdenest(qq6);
qq8: sqrtdenest(qq7); /* yes, three times */
qq9: subst(qq8,qq5,qq4); /* substitute back simplified result */
qq10: factor(qq9);
I don't see how a new user could ever come up with this sequence of
simplifications.
Is there a simpler way?
The kernel of the issue is basically simplifying
log((sqrt(2)+1)*(sqrt(2)-1)^3)/2, which requires knowing about logconcoeffp
and sqrtdenest, which are pretty obscure. The human simplifier would
notice that
((sqrt(2)+1)*(sqrt(2)-1)^3)^(1/2)
= ((sqrt(2)+1)*(sqrt(2)-1)*(sqrt(2)-1)^2) ^ (1/2)
= (1 * (sqrt(2)-1)^2) ^ (1/2)
= sqrt(2)-1
Here's a somewhat harder related challenge: how can you transform this:
((sqrt(a)+1)*(sqrt(a)-1)^3)^(1/2)
or sqrt(a^2-2*a^(3/2)+2*sqrt(a)-1)
to
sqrt(a-1)*(sqrt(a)-1)
(Possibly with some abs's in there.)
-s
On Fri, Apr 5, 2013 at 10:44 AM, Part Marty
<shinabe.munehiro at hotmail.co.jp>wrote:
> Please teach me the modification method of the following formula.
>
> integrate(sqrt(x^2+y^2),x,0,1)$
> integrate(%,y,0,1)$
>
> It leads to the following formula.
>
> 1/3*(sqrt(2)+log(sqrt(2)+1))
>
>
>
> kill(all)$
> algebraic:true$
> lognegint:true$
> assume(y>0)$
> integrate(sqrt(x^2+y^2),x,0,1)$
> integrate(%,y,0,1)$
> logcontract(%);
>
> (%06)(log(-(sqrt(2)-1)/(sqrt(2)+1))+10*asinh(1)-%i*%pi+2^(7/2))/24
>
> _______________________________________________
> Maxima mailing list
> Maxima at math.utexas.edu
> http://www.math.utexas.edu/mailman/listinfo/maxima
>
>