In my paper on geodesics on an ellipsoid, I express the area between a geodesic segment and the equator in terms of an indefinite integral $$\int \frac{t(e'^2) - t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\sigma} \frac{\sin\sigma}2 \,d\sigma,$$ where $$t(x) = x + \sqrt{x^{-1} + 1}\,\sinh^{-1}\!\sqrt x,$$ $e'$ is the second eccentricity, $k = e'\cos\alpha_0$, and $\alpha_0$ is the azimuth of the geodesic when crossing the equator. For oblate ellipsoids, we have $0 < k \le e'$.

In the paper, I evaluate this integral by Taylor expanding the integrand in the limit that $e' \rightarrow 0$. I would like to relax this assumption. I have made some unsystematic (and unsuccessful) stabs at expressing the integral in terms of elliptic integrals.

I would appreciate help with

- expressing the integral in terms of elliptic integrals,
- pointing me to a systematic procedure for doing this,
- proving that the integral can't be expressed in terms of elliptic integrals, or
- expressing the integral in terms of other special functions (especially those which can be numerical evaluated easily).