$\int \left(\frac{\sin \theta }{\cos 3\theta }+\frac{\sin 3\theta }{\cos 9\theta }+\frac{\sin 9\theta }{\cos 27\theta }\right)d\theta isequalto$

$\frac{\sin \theta }{\cos 3\theta }=\frac{2\sin \theta .\cos \theta }{2\cos 3\theta .\cos \theta }=\frac{\sin 2\theta }{2\cos 3\theta .\cos \theta }$

$=\frac{\sin (3\theta -\theta )}{2\cos 3\theta .\cos \theta }=\frac{\sin 3\theta .\cos \theta -\cos 3\theta .\sin \theta }{2\cos 3\theta .\cos \theta }$

$=\frac{1}{2}(\tan 3\theta -\tan \theta )$

$$\begin{gathered} Similarly \frac{\sin 3\theta }{\cos 9\theta }=\frac{1}{2}(\tan 9\theta -\tan 3\theta ) \\ and \frac{\sin 9\theta }{\cos 27\theta }=\frac{1}{2}(\tan 27\theta -\tan 9\theta ) \end{gathered}$$

$$\begin{gathered} \int \left(\frac{\sin \theta }{\cos 3\theta }+\frac{\sin 3\theta }{\cos 9\theta }+\frac{\sin 9\theta }{\cos 27\theta }\right)d\theta \\ =\frac{1}{2}\int (\tan 27\theta -\tan \theta )d\theta \\ =\frac{1}{2}[\frac{1}{27}\log \left|sec27\theta \right|-\log \left|sec\theta \right|]+C \\ =\frac{1}{2}\log \frac{\sqrt[27]{sec27\theta }}{sec\theta }+C \end{gathered}$$