\begin{equation}\int\frac{cos(x)}{1+sin^{2}(x)}dx=\end{equation}
\begin{equation}t=sin(x)\end{equation}
\begin{equation}\frac{dt}{dx}=cos(x)\end{equation}
\begin{equation}dx=\frac{dt}{cos(x)}\end{equation}
\begin{equation}=\int\frac{cos(x)}{1+t^2}\cdot\frac{dt}{cos(x)}=\int\frac{1}{1+t^2}dt=arctg(t)+C=arctg(sinx)+C\end{equation}