I have,
$$ \int uE^{2}\left(u\right)du $$
where $E$ is the complete elliptic integral of the second kind:
$$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} $$
I've tried integrating this by looking for something that differentiates into $E^2(u)$, though can't seem to find anything. Any help?