How to Solve via complexification of $\int \frac{1}{\cos^3x}dx$

Question

How to Solve via complexification of $\int \frac{1}{\cos^3x}dx$

I have just learned complexification for integral solving, and I am trying to test the boundaries of the method. $Re \{e^{i \theta} \} = \cos(x)$. So is it true that $Re \{e^{-3i \theta} \} = \frac{1}{\cos^3(x)}$? Hence the integral I need to solve is:

$$\int \frac{1}{\cos^3x}dx = Re \{ \int e^{-3i\theta}d\theta\}$$

Thanks.

Answer 1

No:

$$\operatorname{Re}\left(e^{-3ix}\right)=\cos (-3)x=\cos 3x.$$

The only way to complexify this one is to write $$\frac{1}{\cos^3 x}=\frac{2^3}{\left(e^{ix}+e^{-ix}\right)^3}$$ and that’s going to be ugly.