The commercial macsyma was able to integrate this, and after some
simplification produces this result:
(((%pi)^4 * l^4)/(sqrt(t - 2 * s + r + 1) * sqrt(t + 2 * s - r + 1) *
sqrt(3 * t - 2 * s - r - 1) * sqrt(3 * t + 2 * s + r - 1) * (t^2 + 2 * t
- r^2 + 1)))
It does not ask any questions involving any of the variables, either.
Dan Stanger
Lin Xie wrote:
> Hi,
> I'm new to Maxima and would like to calculate a full integration of
> %e^(-(x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2-2*r*(x1*x2+x3*x4+x5*x6+x7*x8)-2*s*(x1*x3+x2*x4+x1*x5+x2*x6+x4*x8+x3*x7+x5*x7+x6*x8)-2*t*(x1*x4+x2*x3+x1*x6+x2*x5+x1*x7+x3*x5+x2*x8+x4*x6+x5*x8+x6*x7+x8*x3+x4*x7))/L).
> Here the parameters L, r, s and t satisfy L>0, r^2+s^2+t^2+2*r*s*t<1,
> r^2+s^2<1, r^2+t^2<1 and s^2+t^2<1. The first two
> steps of the integration are just fine. However, I came across a
> problem during carrying out the third integration step. It asks me if
> t^2+s^2-1 is positive or negative! I couldn't figure out why it ask
> this stupid question. Since
> I know that the t^2+s^2-1 is negative, I just answer the question.
> But it asks me another question about the positive or negative,
> because the x1,x2,...,x8 are all variables and I could not define its
> sign. This question almost
> drives me mad.
>
> Here's the execution results:
> (%i1) assume(L>0,r^2<1,s^2<1,t^2<1,r^2+s^2+t^2+2*r*s*t<1,t^2+s^2<1,r^2+s^2<1,r^2+t^2<1);
> integrate(%e^(-(x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2-2*r*(x1*x2+x3*x4+x5*x6+x7*x8)-2*s*(x1*x3+x2*x4+x1*x5+x2*x6+x4*x8+x3*x7+x5*x7+x6*x8)-2*t*(x1*x4+x2*x3+x1*x6+x2*x5+x1*x7+x3*x5+x2*x8+x4*x6+x5*x8+x6*x7+x8*x3+x4*x7))/L),x8,minf,inf);
> (%o1) [L>0,r^2<1,s^2<1,t^2<1,-t^2-2*r*s*t-s^2-r^2+1>0,-t^2-s^2+1>0]
> (%o2) blablabla...
>
> (%i3) integrate(%o2,x1,minf,inf);
> (%o3) blablabla...
>
> (%i4) integrate(%o3,x3,minf,inf);
> Is t^2+s^2-1 positive,negative, or zero? negative;
> Is s(t(2x7+4x6+4x4)+2x7+2rx2)+... positive, negative, or zero? <=
> Here's the problem!
>
> Can anyone give me a help or some suggestions? Thanks!
>
> --
> Lin Xie
> Beijing National Center for Electron Microscopy
> Department of Material Science and Engineering
> Tsinghua University
> Beijing, PR of China
>
>
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