Derivative of complete elliptic integrals, $E(k)$, $K(k)$, etc., are known. But, I don't know about their integrals.
I tried to evaluate the indefinite integral $$\int_0^k K(k)dk\tag1$$ and ended up with $$\int_0^{\arcsin k}\frac{k\phi\cos\phi}{\sqrt{k^2-\sin^2\phi}}d\phi\tag2$$
Is my calculation $(2)$ correct? Is there a closed-form of $(1)$?
Thanks.