$\int \frac{\left({\sin }^3\mathrm{\theta }-{\cos }^3\mathrm{\theta }-{\cos }^2\mathrm{\theta }\right){\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }+{\cos }^2\mathrm{\theta }\right)}^{2007}}{(\sin \mathrm{\theta }{)}^{2009}(\cos \mathrm{\theta }{)}^{2009}}\mathrm{d\theta }=$

$$\begin{gathered} \begin{array}{l}\mathrm{I}=\int \frac{\left({\sin }^3\mathrm{\theta }-{\cos }^3\mathrm{\theta }-{\cos }^2\mathrm{\theta }\right){\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }+{\cos }^2\mathrm{\theta }\right)}^{2007}}{{\sin }^2{\mathrm{\theta cos}}^2{\mathrm{\theta sin}}^{2007}{\mathrm{\theta cos}}^{2007}\mathrm{\theta }}\mathrm{d\theta }\\ =\int \left(\frac{\sin \mathrm{\theta }}{{\cos }^2\mathrm{\theta }}-\frac{\cos \mathrm{\theta }}{{\sin }^2\mathrm{\theta }}-\frac{1}{{\sin }^2\mathrm{\theta }}\right){\left(\frac{1}{\cos \mathrm{\theta }}+\frac{1}{\sin \mathrm{\theta }}+\frac{\cos \mathrm{\theta }}{\sin \mathrm{\theta }}\right)}^{2007}\mathrm{d\theta }\\ \text{ put }\frac{1}{\cos \mathrm{\theta }}+\frac{1}{\sin \mathrm{\theta }}+\frac{\cos \mathrm{\theta }}{\sin \mathrm{\theta }}=t \\ \mathrm{I}=\int t^{2007}\mathrm{dt}=\frac{{\mathrm{t}}^{2008}}{2008}+\mathrm{c}=\frac{1}{2008}{\left(\frac{1}{\cos \mathrm{\theta }}+\frac{1}{\sin \mathrm{\theta }}+\frac{\cos \mathrm{\theta }}{\sin \mathrm{\theta }}\right)}^{2008}+\mathrm{c} \\ =\frac{1}{2008}(\sec \mathrm{\theta }+\mathrm{cosec}\mathrm{\theta }+\cot \mathrm{\theta }{)}^{2008}+\mathrm{c}\end{array} \end{gathered}$$