\int {\frac{1}{{\sin ^2 x + 5\sin x + 6}}dx}  = \int {\frac{1}{{(\sin x + 2)(\sin x + 3)}}dx}  = \left| \begin{array}{l}
 {\mathop{\rm tg}\nolimits} \frac{x}{2} = t \\ 
 \sin x = \frac{{2t}}{{1 + t^2 }} \\ 
 dx = \frac{{2dt}}{{1 + t^2 }} \\ 
 \end{array} \right| = \int {\frac{{\frac{{2dt}}{{1 + t^2 }}}}{{(\frac{{2t}}{{1 + t^2 }} + 2)(\frac{{2t}}{{1 + t^2 }} + 3)}}}  = \int {\frac{{\frac{{2dt}}{{1 + t^2 }}}}{{(\frac{{2t^2  + 2t + 2}}{{1 + t^2 }})(\frac{{3t^2  + 2t + 3}}{{1 + t^2 }})}}}  = \int {\frac{{1 + t^2 }}{{(t^2  + t + 1)(3t^2  + 2t + 3)}}dt}  = \int {\left[ {\frac{1}{{t^2  + t + 1}} - \frac{2}{{3t^2  + 2t + 3}}} \right]dt}  = ........