While working on an engineering problem I needed to calculate this integral, so I typed it into the Wolfram Alpha and got this:
If you look at the logarithms, you will see that one of arguments have the opposite sign than the other. This means logarithms with negative arguments there, which means the result will definitely have imaginary terms. Why?
It doesn't look like there is a way to cancel the imaginary terms when I calculate a definite integral, but I think it's definitely possible to set the parameters and the range so the integrand remains real: large $a$, large $c$, small $b$ and the range of $x$ is around 0.