Simple Integral Involving the Square of the Elliptic Integral

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?

• Do you specifically need an anti-derivative? Or would it suffice to obtain the definite integral over some interval? – David H Aug 28 '14 at 23:04