When I ask maxima to
integrate(x/((x+dx*r)^2+(y+dy*r)^2+(z+dz*r)^2)^(3/2),r); it does give a
result that could be easily (even I could do it) simplified a lot (see
output below).
I notice that for (slightly) simpler integrals, maxima does simplify the
result, so it seems it has hit a max simplification-search-depth somewhere,
given up, and dumped the not-simplified result.
Is there a variable I can set so that maxima will try harder to simplify?
(at the moment, the results comes just about instantly).
(%i133) integrate(x/((x+rx*r)^2+(y+ry*r)^2+(z+rz*r)^2)^(3/2),r);
2 2 2 2 2 2 2
Is (2 rz z + 2 ry y + 2 rx x) - 4 (rz + ry + rx ) (z + y + x )
zero or
nonzero?
n;
2 2 2
(%o133) x (4 rz z/(sqrt(z + r (2 rz z + 2 ry y + 2 rx x) + y + x
2 2 2 2 2 2 2 2 2 2
+ r (rz + ry + rx )) (4 (rz + ry + rx ) (z + y + x )
2
- (2 rz z + 2 ry y + 2 rx x) )) + 4 ry y
2 2 2 2 2 2 2
/(sqrt(z + r (2 rz z + 2 ry y + 2 rx x) + y + x + r (rz + ry + rx ))
2 2 2 2 2 2 2
(4 (rz + ry + rx ) (z + y + x ) - (2 rz z + 2 ry y + 2 rx x) ))
2 2 2
+ 4 rx x/(sqrt(z + r (2 rz z + 2 ry y + 2 rx x) + y + x
2 2 2 2 2 2 2 2 2 2
+ r (rz + ry + rx )) (4 (rz + ry + rx ) (z + y + x )
2 2 2 2
- (2 rz z + 2 ry y + 2 rx x) )) + 4 r (rz + ry + rx )
2 2 2 2 2 2 2
/(sqrt(z + r (2 rz z + 2 ry y + 2 rx x) + y + x + r (rz + ry + rx ))
2 2 2 2 2 2 2
(4 (rz + ry + rx ) (z + y + x ) - (2 rz z + 2 ry y + 2 rx x) )))
(%i134) f90(%);
x*(4*rz*z/ &
(sqrt(z**2+r*(2*rz*z+2*ry*y+2*rx*x)+y**2+x**2+r**2*(rz** &
2+ry**2+rx**2))*(4*(rz**2+ry**2+rx**2)*(z**2+y**2+x**2)-(2* &
rz*z+2*ry*y+2*rx*x)**2))+4*ry*y/ &
(sqrt(z**2+r*(2*rz*z+2*ry*y+2*rx*x)+y**2+x**2+r**2*(rz** &
2+ry**2+rx**2))*(4*(rz**2+ry**2+rx**2)*(z**2+y**2+x**2)-(2* &
rz*z+2*ry*y+2*rx*x)**2))+4*rx*x/ &
(sqrt(z**2+r*(2*rz*z+2*ry*y+2*rx*x)+y**2+x**2+r**2*(rz** &
2+ry**2+rx**2))*(4*(rz**2+ry**2+rx**2)*(z**2+y**2+x**2)-(2* &
rz*z+2*ry*y+2*rx*x)**2))+4*r*(rz**2+ry**2+rx**2)/ &
(sqrt(z**2+r*(2*rz*z+2*ry*y+2*rx*x)+y**2+x**2+r**2*(rz** &
2+ry**2+rx**2))*(4*(rz**2+ry**2+rx**2)*(z**2+y**2+x**2)-(2* &
rz*z+2*ry*y+2*rx*x)**2)))
(I show the f90() output for those that read with proportional font).
In the result, this is repeated 4 times:
(sqrt(z**2+r*(2*rz*z+2*ry*y+2*rx*x)+y**2+x**2+r**2*(rz**2+ry**2+rx**2))*
(4*(rz**2+ry**2+rx**2)*(z**2+y**2+x**2)-(2*rz*z+2*ry*y+2*rx*x)**2))
So, the whole result can (at least) be simplified as:
4*x*(rx*x+ry*y+rz*z + r*(rx**2+ry**2+rz**2))/
(sqrt(z**2+r*(2*dz*z+2*dy*y+2*dx*x)+y**2+x**2+(dz**2+dy** &
2+dx**2)*r**2)*(4*(dz**2+dy**2+dx**2)*(z**2+y**2+x**2)-(2* &
dz*z+2*dy*y+2*dx*x)**2))
Thanks,
joost witteveen