How do I find the answer to complete elliptic integral of the second kind without computer (i.e. with paper and pen)?

Question

I know that $$E(\theta|k^2)=\int_0^\theta{\sqrt{1-k^2\sin^2\phi}}\,d\phi.$$

I want to find $E(k)$ where $k=0.36$.

With a website I computed it to be approximately $1.518$.

How can I find the answer without computer?