TITLE: Reduction of integral for geodesic area to elliptic integrals
QUESTION [5 upvotes]: 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).

REPLY [2 votes]: I have seen your paper. You have worked out a series with $e'\rightarrow 0$. You can do the same on the other side $e'\rightarrow\infty$. Also for this series the integrals can be evaluated. In this way, you will be able to get a satisfactory numerical evaluation of the integral for a wide range of values of $e'$. That this can be done can be easily seen by noting the asymptotic series
$$
  t(x)=x+\ln 2+\frac{1}{2}\ln x+\left(\frac{1}{2}\ln 2+\ln x+\frac{1}{4}\right)\frac{1}{x}+\ldots
$$
that entails both logarithm and power terms. For example, for your integral you will get
$$
\int\frac{t(e'^2)-t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\sigma}\frac{\sin\sigma}{2}d\sigma=
-\frac{1}{2}\cos\sigma+O\left(\frac{1}{e'^2}\right)
$$
and you can evaluate the higher order terms by any computer based program you prefer.