f(x):=csc(2*x)*(sqrt(sin(x))-sqrt(cos(x)))^4/((1-csc(2*x))*(cos(x)-sin(x))^2);

    Machine Integrate f(x) to obtain F(x)

F(x):= (5*sin(x)^2 - 4*sqrt(sin(x))*cos(x)^(3/2) + 6*sin(x)^(3/2)*sqrt(cos(x)) - 3*sin(x)*cos(x) - 2)/
       (3*(sqrt(cos(x))-sqrt(sin(x)))*(sqrt(sin(x))+sqrt(cos(x)))^3) + C;

fs(x):=trigsimp(f(x))$ fs(x);

Fs(x):=trigsimp(F(x))$ Fs(x);

Fs(%pi/2)-Fs(0); ratsimp(%);