i have know idea on how to do this problem:
int csc^{2}(2t)} / sqrt{1 + cot(2t)} dt
Is the integrand not very closly related to the derivative of sqrt(1+cot(2t)) ?
In which case the Fundamental Theorem of Calculus will give the integral.
Unless I have made some elemenatry mistake:
d/dt sqrt(1+cot(2t))=(1/2) 2 (-csc^2(2t))/sqrt(1+cot(2t))=-csc^2(2t)/sqrt(1+cot(2t))
so:
int csc^{2}(2t)} / sqrt{1 + cot(2t)} dt = -sqrt(1+cot(2t))
RonL