all the sin & cos terms are in denomenater .

\int\frac{1}{(sin\theta-2cos\theta)(2sin\theta+cos\theta)}d\theta

Sometimes these tough ones can be done by using obscure-looking identities. Rewrite as:

***\frac{-2}{5}\int\frac{1}{cos(2\theta-tan^{-1}(\frac{3}{4}))}d\theta

Let u=2\theta-tan^{-1}(\frac{3}{4}), \;\ \theta=\frac{u+tan^{-1}(\frac{3}{4})}{2}, \;\ d\theta=\frac{du}{2}

Making the subs, the integral becomes:

\frac{-1}{5}\int sec(u)du

=\frac{-1}{5}ln(sec(u)+tan(u))

Resub:

\frac{-1}{5}ln\left[sec\left(2\theta-tan^{-1}\left(\frac{3}{4}\right)\right)+tan\left(2\theta-tan^{-1}\left(\frac{3}{4}\right)\right)\right]+C

***This can be derived by using the subtraction formula for cosine.

By the way, \theta is spelled 'theta', not thita.