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Evaluating

$$ \int \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2} \ dx$$

should just give $\frac{f(x)}{g(x)}$. Now I have a similar quotient over $\mathbb{C}$, at least it looks similar. It's of the form

$$ \int \frac{ \left(f(x) \bar f'(x)- f'(x)\bar f(x) + g(x)\bar g'(x) - g'(x)\bar g(x) \right)}{f^2(x) + g^2(x)} \ dx$$

Is there a known solution to this type of integrand? It seems so related that I thought it must be solvable analytically.

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