I've run into the following integral:

$\int \frac{K(k)}{k} dk$

where $K$ is the complete elliptic integral of the first kind

$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin\theta}}$.

I've looked in Byrd, Friedman, "Handbook of Elliptic Integrals...", and found that $\int K/k dk = \int E/k dk - E$ (where $E$ is the complete elliptic integral of the second kind). There are some other similar formulas, too, such as $\int K/k^2 dk = -E/k$.

This leads me to suspect that there is no "nice" formula for $\int \frac{K(k)}{k} dk$. Is there a sense in which I can make this precise?