A question about the derivative of $\arctan(f(x))$

Question

We all know that:

$$ \int{\frac{f'(x)}{(f(x))^2 + 1}} dx = \arctan(f(x)) + c.$$

But what happens if we change the $1$ in the denominator? For example: $$\int{\frac{f'(x)}{(f(x))^2 + c}} dx. \qquad (c \in \mathbb R)$$

Answer 1

Then you can use a function $g(x) = \frac{f(x)}{\sqrt{c}}$, $(c>0)$ and everything will turn out nicely ...