I'm looking for an algebraic addition formula for a class of nonstandard elliptic integrals.
\begin{align*} r&=\frac{\sqrt{\left(1+pu\right)\left(1+u\right)\left(1-v\right)\left(1-qv\right)\,\,}+\sqrt{\left(1+pv\right)\left(1+v\right)\left(1-u\right)\left(1-qu\right)\,\,}}{2+p+q+\left(p-q\right)\left(u+v\right)-\left(p+q+2pq\right)uv}\\ s&=\frac{\sqrt{\left(1-\left(p+1\right)\left(q+1\right)r^{2}\right)\left(1-2\left(p+q\right)r^{2}\right)}-\left(1-p\right)r}{1-\left(p+q\right)\left(p+1\right)r^{2}}\\ w&=\frac{1-s^{2}}{ps^{2}-1} \end{align*}
\begin{align*} &\int_{0}^{w}\frac{1}{\sqrt{\left(1+pt\right)\left(1+t\right)\left(1-t\right)\left(1-qt\right)}}\mathrm{d}t\\ &\qquad=\int_{0}^{u}\frac{1}{\sqrt{\left(1+pt\right)\left(1+t\right)\left(1-t\right)\left(1-qt\right)}}\mathrm{d}t\\ &\qquad\qquad\quad+\int_{0}^{v}\frac{1}{\sqrt{\left(1+pt\right)\left(1+t\right)\left(1-t\right)\left(1-qt\right)}}\mathrm{d}t\\ \end{align*}
My question is whether I can find a more concise form of $w$
@Paramanand Singh
We can find the degenerate case, namely, $p=q=k$
\begin{align*} w=\dfrac{u\sqrt{(1-v^2)(1-k^2v^2)\,}+v\sqrt{(1-u^2)(1-k^2u^2)\,}}{1-k^2u^2v^2} \end{align*}
\begin{align*} &\int_{0}^{w}\frac{1}{\sqrt{\left(1+kt\right)\left(1+t\right)\left(1-t\right)\left(1-kt\right)}}\mathrm{d}t\\ &\qquad=\int_{0}^{u}\frac{1}{\sqrt{\left(1+kt\right)\left(1+t\right)\left(1-t\right)\left(1-kt\right)}}\mathrm{d}t\\ &\qquad\qquad\quad+\int_{0}^{v}\frac{1}{\sqrt{\left(1+kt\right)\left(1+t\right)\left(1-t\right)\left(1-kt\right)}}\mathrm{d}t\\ \end{align*}