Trasformation of elliptic integral $\Pi(n;x|m)$ in function of $F(x|m)$ and $E(x|m)$

Question

I everyone,
I calculated these two formulas for $m<1$ and $z\in\mathbb{C}$: $$K(m)=F\left(\left.\frac{\pi}{2}-z\right|m\right)+\frac{1}{\sqrt{1-m}}\cdot F\left(z\left|\frac{m}{m-1}\right.\right)$$ $$E(m)=E\left(\left.\frac{\pi}{2}-z\right|m\right)+\sqrt{1-m}\cdot E\left(z\left|\frac{m}{m-1}\right.\right)$$ Where

1. $K(m)$ and $F(x|m)$ are the complete and incomplete elliptic integrals of $1°$ kind
2. $E(m)$ and $E(x|m)$ are the complete and incomplete elliptic integrals of $2°$ kind
3. $\Pi(n|m)$ and $\Pi(n;x|m)$ are the complete and incomplete elliptic integrals of $3°$ kind

Defined as in this link: https://functions.wolfram.com/EllipticIntegrals/

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Question

I would like to extend these formulas for the complete and incomplete elliptic integral of $3^°$ kind

Is there some formula to express $\Pi(n;x|m)$ in function of $F(x|m)$ and $E(z|m)$ or $\Pi(n|m)$ in function of $K(m)$ and $E(m)$?

Answer 1

$\small\Pi(n,z|m)=\int_0^z\frac{1}{(1-n\sin^2\phi)\sqrt{1-m\sin^2\phi}}d\phi.$

To complete add $\small\int_z^{\frac\pi 2}\frac{1}{(1-n\sin^2\phi)\sqrt{1-m\sin^2\phi}}d\phi \stackrel{\phi\to \frac\pi 2-\phi}{=}\frac{1}{(1-n)\sqrt{1-m}}\int_0^{\frac\pi 2-z}\frac{1}{(1-\frac{n}{n-1}\sin^2\phi)\sqrt{1-\frac{m}{m-1}\sin^2\phi}}d\phi.$

So,

$\Pi(n,m)=\Pi(n,z|m)+\frac{1}{(1-n)\sqrt{1-m}}\Pi(\frac n{n-1},\frac\pi 2-z|\frac m{m-1}).$