Finding the indefinite integral of a function that is only real over a particular domain

Question

I am having trouble evaluating this integral whose solution is a polar equation. $$\theta=\int \sqrt {\frac{-c^2}{r^2(r+c^2)}} dr $$ Here, $c$ is a constant. Using the substitution $u=\sqrt {r+c^2}$, I was able to obtain the solution: $$\theta=i\ln{\left|\frac{\sqrt {r+c^2}-c}{\sqrt {r+c^2}+c}\right|}+D$$ The expression being integrated is real for $r<-c^2$. Two points on the curve are $(-2053,0)$ and $(-415.8,\pi)$. In both cases $r$ is negative and so I think it is possible that a real solution exists for a limited domain. However, I am unsure about how a real solution can be obtained from the solution I got. How can this be solved? Can a real polar curve be generated? Is there an alternate solution to the integration that can be of use?