integrate((x^4+x^2+4)/(x^6-7*x^4+14*x^2+1),x,-2,2);
gives $0.$
From reading "Improving Exact Integrals From Symbolic Algebra Systems" by Fateman and Kahan, page 5,
this should be $2\pi.$
Raymond Toy
2010-04-19
Maxima converts this integral to an integral from 0 to inf. This new integral is evaluted using resides, and maxima fails to find the roots of the denominator:
9*x^6-42*x^5+1031*x^4-1932*x^3+1031*x^2-42*x+9
Maxima erroneously returns 0 in this case.
Aleksas Domarkas
2010-04-19
For integrating we define block "int_rac(f,x)":
(%i1) int_rac(f,x):=block('integrate(f,x),map(gfactor,%%),
ev(%%,nouns),rectform(%%),logcontract(%%))$
1 example
(%i2) integrate((x^4+x^2+4)/(x^6-7*x^4+14*x^2+1),x);
(%o2) integrate((x^4+x^2+4)/(x^6-7*x^4+14*x^2+1),x)
(%i3) f:first(%);
(%o3) (x^4+x^2+4)/(x^6-7*x^4+14*x^2+1)
We find antiderivative F :
(%i4) F:int_rac(f,x);
(%o4) -atan2(x^2-1,x^3-4*x)
Test:
(%i5) diff(F,x),ratsimp;
(%o5) (x^4+x^2+4)/(x^6-7*x^4+14*x^2+1)
(%i6) limit(F,x,1,minus);
(%o6) %pi
(%i7) limit(F,x,1,plus);
(%o7) -%pi
Then F is discontinous at x=1.
(%i8) wxplot2d([f,F], [x,-5,5],[y,-5,5])$
(%t8) << Graphics >>
Function f is even. Then
(%i13) 'integrate(f,x,-2,2)=2*(ev(F,x=0)-ev(F,x=-2));
(%o13) integrate((x^4+x^2+4)/(x^6-7*x^4+14*x^2+1),x,-2,2)=2*%pi
2 example
(%i10) S:integrate(1/(x^4+6*x^2+1),x);
(%o10) integrate(1/(x^4+6*x^2+1),x)
(%i11) sol:int_rac(1/(x^4+6*x^2+1),x);
(%o11) (2^(5/2)*atan((2*x)/(x^2+1))+4*atan(x/(sqrt(2)+1))+4*atan(x/(sqrt(2)-1)))/2^(9/2)
Test:
(%i12) diff(%,x),ratsimp;
(%o12) 1/(x^4+6*x^2+1)
Raymond Toy
2010-04-29
Raymond Toy
2010-04-29
The bug in keyhole integration has been fixed. (defint.lisp, rev 1.76) This now returns a noun form.
Better than returning 0, but not as good as returning 2*%pi.
Setting to pending/fixed.
SourceForge Robot
2010-05-13
SourceForge Robot
2010-05-13
This Tracker item was closed automatically by the system. It was
previously set to a Pending status, and the original submitter
did not respond within 14 days (the time period specified by
the administrator of this Tracker).