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Sage 6.5:

sage: (A,t) = var('A,t') sage: assume(A > 0) sage: assume(A < 1/3) sage: num = A^2 * cos(t)^2 + 2abs(A cos(2t)) sage: denom = 2 + A^2cos(t)^2 + 2abs(Acos(2t)) sage: beta = -num/denom sage: integral(beta,t,0,2pi) -2*pi

This can't be right as beta has absolute value less than 1 so its integral over 0 to 2 pi has to have absolute value less than 2 pi.

Sage 6.5:

sage: (A,t) = var('A,t')
sage: assume(A > 0)
sage: assume(A < 1/3)
sage: num = A^2 * cos(t)^2 + 2abs(A cos(2t))
2*abs(A* cos(2*t))
sage: denom = 2 + A^2cos(t)^2 A^2*cos(t)^2 + 2abs(Acos(2t))
2*abs(A*cos(2*t))
sage: beta = -num/denom
sage: integral(beta,t,0,2pi)
-2*pi
integral(beta,t,0,2*pi)
-2*pi

Sage 6.5:

sage: (A,t) = var('A,t')
sage: assume(A > 0)
sage: assume(A < 1/3)
sage: num = A^2 * cos(t)^2 + 2*abs(A* cos(2*t))
sage: denom = 2 + A^2*cos(t)^2 + 2*abs(A*cos(2*t))
sage: beta = -num/denom
sage: integral(beta,t,0,2*pi)
-2*pi

This can't be right as beta has absolute value less than 1 so its integral over 0 to 2 pi has to have absolute value less than 2 pi.

Sage 6.5:

sage: (A,t) = var('A,t')
sage: assume(A > 0)
sage: assume(A < 1/3)
sage: num = A^2 * cos(t)^2 + 2*abs(A* cos(2*t))
sage: denom = 2 + A^2*cos(t)^2 + 2*abs(A*cos(2*t))
sage: beta = -num/denom
sage: integral(beta,t,0,2*pi)
-2*pi

This can't be right as beta has absolute value less than 1 so its integral over 0 to 2 pi has to have absolute value less than 2 pi.