Any evil will be brought to justice! $$ \int_{0 }^{\sqrt{\frac23}} \frac{(K(x)-E(x))K\left (\frac{x\sqrt{2-3x^2} }{x^2-1} \right ) }{ x\sqrt{1-x^2} }\text{d} x-\frac13 \int_{\sqrt{\frac23}}^1 \frac{\sqrt{1-x^2} (K(x)-E(x))K\left (\frac{x\sqrt{3x^2-2} }{2x^2-1} \right ) }{ x(2x^2-1) }\text{d} x=\frac{\pi^2}{12}. $$