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It doesn't help resolving the question immediately, but it seems the problem here is maxima's "abs_integrate" package. In maxima:
(%i1) display2d:false;
(%o1) false
(%i2) integrate(sqrt(16*r^2*sin(t)^4 + 9*r^4 - 16*r^2*sin(t)^2 + 10*r^2 + 1)/(r^2 + 1),t,0,2*%pi);
(%o2) ('integrate(sqrt(16*r^2*sin(t)^4-16*r^2*sin(t)^2+9*r^4+10*r^2+1),t,0,2*%pi)) /(r^2+1)
(%i3) load(abs_integrate);
(%o3) "/usr/local/sage/sage-git/local/share/maxima/5.34.1/share/contrib/integration/abs_integrate.mac"
(%i4) integrate(sqrt(16*r^2*sin(t)^4 + 9*r^4 - 16*r^2*sin(t)^2 + 10*r^2 + 1)/(r^2 + 1),t,0,2*%pi);
sign: argument cannot be imaginary; found %i
#0: intfudu(exp=sqrt(9*r^4*%e^-(4*%i*t)+8*r^2*%e^-(4*%i*t)+%e^-(4*%i*t)+r^2*%e^-(8*%i*t)+r^2)*%e^(2*%i*t),%voi=t (partition.mac line 95)
#1: extra_integrate(q=sqrt(9*r^4*%e^-(4*%i*t)+8*r^2*%e^-(4*%i*t)+%e^-(4*%i*t)+r^2*%e^(8*%i*t)+r^2)*%e^(2*%i*t),x=t)
-- an error. To debug this try: debugmode(true);
so, it doesn't seem within maxima's current capabilities to express this integral into functions that allow for more efficient approximation methods.
