I've seen these two functions often appear interchangeably in differential equation and integral solutions. There seems to be an intuitive relation between those two.
One can also notice their similarity on the basic composite function integrals:
$$\int \frac{f(x)f'(x)}{f(x)^2+1} dx = \frac12 \ln\left|f^2(x)+1\right| +C \quad (1)$$ $$\int \frac{f'(x)}{f(x)^2+1} dx = \arctan\left|f(x)\right| +C\quad (2)$$
My question is:
Is there a way to write $\arctan(f(x)) = \ln(g(x))$, with the restriction that $f,g$ are real functions?.
In other words, is there a way to reduce $\arctan$ functions to $\ln$'s and visa-versa?
The reason why they often appear next to each other is
$$\int\frac{dz}z=\log(z)=\log(\sqrt{x^2+y^2})+i\arctan\left(\frac yx\right).$$
This integral is at the very heart of the calculus of residues and is ubiquitous in the antiderivative of rational functions.