∫1/sin2x dx = ∫cosec2x dx = 1/2 log[cosec2x - cot2x] + c = 1/2 log[tan x] + c
Detailed derivation of
∫cosec x dx = ∫cosec x(cosec x - cot x)/(cosec x - cot x) dx
= ∫(cosec2x - cosecxcotx)/(cosecx - cotx) dx
= log[cosecx - cotx] = log[(1-cosx)/sinx]
= log[2sin2(x/2)/2sin(x/2)cos(x/2)]
= log[tan(x/2)]