For $\left(\varphi,\nu,\kappa\right)\in\mathbb{R}^{3}$ such that $-\frac{\pi}{2}\le\varphi\le\frac{\pi}{2}\land-1<\kappa<1\land0<1-\nu\sin^{2}{\left(\varphi\right)}$, the incomplete elliptic integrals of the first/second/third kind may be given, respectively, by
$$F{\left(\varphi,\kappa\right)}:=\int_{0}^{\sin{\left(\varphi\right)}}\mathrm{d}x\,\frac{1}{\sqrt{\left(1-x^{2}\right)\left(1-\kappa^{2}x^{2}\right)}},$$
$$E{\left(\varphi,\kappa\right)}:=\int_{0}^{\sin{\left(\varphi\right)}}\mathrm{d}x\,\frac{\sqrt{1-\kappa^{2}x^{2}}}{\sqrt{1-x^{2}}},$$
$$\Pi{\left(\varphi,\nu,\kappa\right)}:=\int_{0}^{\sin{\left(\varphi\right)}}\mathrm{d}x\,\frac{1}{\left(1-\nu x^{2}\right)\sqrt{\left(1-x^{2}\right)\left(1-\kappa^{2}x^{2}\right)}}.$$
Also, define the auxiliary function $R_{C}:\mathbb{R}_{\ge0}\times\mathbb{R}_{>0}\rightarrow\mathbb{R}$ by the improper integral
$$R_{C}{\left(x,y\right)}:=\frac12\int_{0}^{\infty}\mathrm{d}t\,\frac{1}{\left(t+y\right)\sqrt{t+x}};~~~\small{x\in\mathbb{R}_{\ge0}\land y\in\mathbb{R}_{>0}}.$$
The incomplete elliptic integral of the third kind $\Pi$ has three connection formulas that allow for changing the elliptic characteristic while keeping the modulus and amplitude the same.
Suppose $\left(\varphi,\nu,\kappa\right)\in\mathbb{R}^{3}\land0<\varphi<\frac{\pi}{2}\land0<\kappa<1\land\nu<\csc^{2}{\left(\varphi\right)}$, and set $c:=\csc^{2}{\left(\varphi\right)}$.
Connection Formula $1$: Assuming $\neg\left[0\le\nu\le\frac{\kappa^{2}}{c}\right]$,
$$\Pi{\left(\varphi,\nu,\kappa\right)}+\Pi{\left(\varphi,\rho,\kappa\right)}=F{\left(\varphi,\kappa\right)}+\sqrt{c}\,R_{C}{\left((c-1)(c-\kappa^{2}),(c-\nu)(c-\rho)\right)},$$
where $\nu\rho=\kappa^{2}$.
Connection Formula $2$: Assuming $\nu<1$,
$$\left(\kappa^{2}-\nu\right)\Pi{\left(\varphi,\nu,\kappa\right)}+\left(\kappa^{2}-\rho\right)\Pi{\left(\varphi,\rho,\kappa\right)}=\kappa^{2}F{\left(\varphi,\kappa\right)}-\nu\rho\sqrt{c-1}\,R_{C}{\left(c(c-\kappa^{2}),(c-\nu)(c-\rho)\right)},$$
where $\left(1-\nu\right)\left(1-\rho\right)=1-\kappa^{2}$.
Connection Formula $3$: Assuming $\neg\left[\frac{\kappa^{2}\cos^{2}{\left(\varphi\right)}}{1-\kappa^{2}\sin^{2}{\left(\varphi\right)}}\le\nu\le\kappa^{2}\right]$,
$$\left(1-\nu\right)\Pi{\left(\varphi,\nu,\kappa\right)}+\left(1-\rho\right)\Pi{\left(\varphi,\rho,\kappa\right)}=F{\left(\varphi,\kappa\right)}+\left(1-\nu-\rho\right)\sqrt{c-\kappa^{2}}\,R_{C}{\left(c(c-1),(c-\nu)(c-\rho)\right)},$$
where $\left(\kappa^{2}-\nu\right)\left(\kappa^{2}-\rho\right)=\kappa^{2}\left(\kappa^{2}-1\right)$.
It seems to me it should be possible to derive these connection formulas directly from the integral representations of $F$, $\Pi$, and $R_{C}$, but such a proof has eluded me. Any guidance proving one or all three of these formulas would be greatly appreciated.