I need to find
$\int ^{x}_{0} \alpha^{2}\sin^{2}\left( \dfrac {2 \pi x}{w}\right) dL$
where
$L=\dfrac {2kE\left( \dfrac {2\pi \alpha }{k}\right) }{\pi}\approx \dfrac {k+w}{2}\left( \dfrac {3\left( k-w\right)^{2}}{\left( k+w\right) \left( 10k+10w+\sqrt {k^{2}+w^{2}+14kw}\right) }+1\right)$
$k=\sqrt {4\pi ^{2}\alpha ^{2}+w^{2}}$
and
$E\left( m\right)=\int^{\dfrac {\pi}{2}}_{0}\sqrt {1-m^{2}\sin ^{2}\theta }d\theta$.
I have tried approximating $L$ and solving for $\alpha$ and $w$ in order to substitue into the first equation but it seems for the approximation to maintain a reasonable level of accuracy, the solutions for $\alpha$ and $w$ must be horrendously long.
I would greatly appreciate any input.