Can $\int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2-x^2})}e^{ikx}dx$ be found in closed form?

Question

I am trying to see if it is worth pursuing to try to calculate the following integral analytically:

\begin{align} \int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2 -x^2})}e^{ikx}dx\end{align}

If the singularities are problematic, these can be removed by introducing a regularisation parameter $\epsilon > 0.25$:

\begin{align} \int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2+\epsilon^2b^2 -x^2})}e^{ikx}dx \end{align}