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?



Integrating the complete elliptic integral K
Wolfram is your friend. Mathematica comes up with
$${\pi x\over 8} {}\_4F\_3(1,1,3/2,3/2;2,2,2;x)+{\pi\over 2}\log(x).$$


