calculus - Integration by Euler's formula

How do you integrate the following by using Euler's formula, without using integration by parts?

$$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}d\theta$$

I did integrate it by parts, by writing the $3$ in the numerator as $3\sin^2 {\theta}+3\cos^2{\theta}$, and then splitting the numerator.

But can it be solved by using complex numbers and the Euler's formula?

Answer

Hint

When you have an expression with a squared denominator, you could think that the solution is of the form

$$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}~d\theta=\frac{a+b\sin \theta+c\cos \theta}{3\cos {\theta}+4}$$ Differentiate the rhs and identify terms. You will get very simple results.