kill(all)$

van=facebook.com/LearningCalculus$

f(x):=(sqrt(tan(x))-sqrt(tan(2*x)))/(sqrt(tan(2*x))+sqrt(tan(x)));

F(x) = integrate('(f(x)),x);

 F(x) = (
  -4*tan(x)^(3/2) -
   4*sqrt(tan(2*x))*tan(x) -
   4*sqrt(tan(x))*(2*x + sin(3*x)*sec(x)) +
     sqrt(tan(2*x))*(sqrt(2)*sqrt(tan(x)^2 - 1)*(log(-3*tan(x)^2 -
   2*sqrt(2)*sqrt(tan(x)^2 - 1)*tan(x) + 1) -
     log(-3*tan(x)^2 + 2*sqrt(2)*sqrt(tan(x)^2 - 1)*tan(x) + 1)) +
   4*sin(3*x)*sec(x))
 )/(8*sqrt(tan(x))) + C;