Помогите, пожалуйста, найти интеграл
int sin x/(1 + sin x)^2 dx
Заранее спасибо!

int sin x/(1 + sin x)^2 dx =
= | tg (x/2) = t => sin x = 2 * t/(1 + t^2), x = 2 * arctg t, dx = 2/(1 + t^2) dt | =
= int (2 * t/(1 + t^2))/(1 + 2 * t/(1 + t^2))^2 * 2/(1 + t^2) dt =
= int 4 * t/(1 + 2 * t/(1 + t^2))^2 * 1/(1 + t^2)^2 dt =
= int 4 * t/((1 + t^2) + 2 * t)^2 dt = int 4 * t/(t^2 + 2 * t + 1)^2 dt =
= 4 * int (t + 1 - 1)/(t + 1)^4 dt = 4 * int (t + 1)/(t + 1)^4 dt - 4 * int dt/(t + 1)^4 =
= 4 * int dt/(t + 1)^3 - 4 * int dt/(t + 1)^4 =
= 4 * int (t + 1)^(-3) dt - 4 * int (t + 1)^(-4) dt =
= 4 * 1/(-3 + 1) * (t + 1)^(-3 + 1) - 4 * 1/(-4 + 1) * (t + 1)^(-4 + 1) + C =
= -2 * (t + 1)^(-2) + 4/3 * (t + 1)^(-3) + C = -2/(t + 1)^2 + 4/3 * 1/(t + 1)^3 + C =
= | t = tg (x/2) | = -2/(tg (x/2) + 1)^2 + 4/3 * 1/(tg (x/2) + 1)^3 + C

Спасибо большое