According to Abel's astonishing paper "RECHERCHES SUR LES FONCTIONS ELLIPTIQUES.[1]", we can get \begin{align*} \varphi_3(x,\kappa_3)&=\frac{x\cdot\left(\sqrt{3}-\,\kappa_3x^{2}\right)}{1+\sqrt{3}\,\kappa_3x^{2}}\\ \int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1+\kappa_3^{2}t^{2}\right)}} &=\dfrac{1}{\sqrt{3}}\int_{0}^{\varphi_3(x,\kappa_3)}\frac{\mathrm{d}t}{\sqrt{\left(1+t^{2}\right)\left(1-\kappa_3^{2}t^{2}\right)}}\\ \Big(\left|x\right|&\leqslant2-\sqrt{3}\Big) \end{align*} and \begin{align*} \varphi_5(x,\kappa_5)&=\frac{x\cdot\left(\sqrt{5}-\sqrt{10+10\sqrt{5}}\,\kappa_5x^{2}+\,\kappa_5^{2}x^{4}\right)}{1+\sqrt{10+10\sqrt{5}}\,\kappa_5x^{2}+\sqrt{5}\,\kappa_5^{2}x^{4}}\\ \int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1+\kappa_5^{2}t^{2}\right)}} &=\dfrac{1}{\sqrt{5}}\int_{0}^{\varphi_5(x,\kappa_5)}\frac{\mathrm{d}t}{\sqrt{\left(1+t^{2}\right)\left(1-\kappa_5^{2}t^{2}\right)}}\\ \Big(\left|x\right|&\leqslant\,l_5\Big), \end{align*} where \begin{align*} \kappa_1&=1=\frac{\sqrt{1-k_1^2}}{k_1}\\ \kappa_3&=2+\sqrt{3}=\frac{\sqrt{1-k_3^2}}{k_3}\\ \kappa_5&=2+\sqrt{5}+2\sqrt{2+\sqrt{5}}=\frac{\sqrt{1-k_5^2}}{k_5} \end{align*}
\begin{align*} l_5&=\sqrt{\frac{1-\sqrt{1-\frac{\sqrt{10+10\sqrt{5}}-\sqrt{\sqrt{5}}-\sqrt{5\sqrt{5}}}{2\left(2+\sqrt{5}+2\sqrt{2+\sqrt{5}}\right)}}}{1+\sqrt{1+\frac{\sqrt{10+10\sqrt{5}}-\sqrt{\sqrt{5}}-\sqrt{5\sqrt{5}}}{2\left(2+\sqrt{5}-2\sqrt{2+\sqrt{5}}\right)}}}}\\ &=\frac{1}{2} \left(2+\sqrt{5}+\sqrt{25+10 \sqrt{5}}-\sqrt{30+14 \sqrt{5}+2 \sqrt{425+190 \sqrt{5}}}\right) \end{align*} It is worth noting that these moduli are related to the singular values of elliptic integrals[2].
My confusion is as follows, whether there is an algebraic function $\varphi_2(x,\kappa_2)$ (rational function, irrational function can be) and the corresponding modulus, satisfying the following equation? \begin{align*} \int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1+\kappa_2^{2}t^{2}\right)}} =\dfrac{1}{\sqrt{2}}\int_{0}^{\varphi_2(x,\kappa_2)}\frac{\mathrm{d}t} {\sqrt{\left(1+t^{2}\right)\left(1-\kappa_2^{2}t^{2}\right)}} \end{align*} If it exists, how to find it?
[1] Abel N.-H. (1828) Recherches sur les fonctions elliptiques (suite du Mémoire n°9). Cr. [Journal für die reine und angewandte Mathematik. Journal de Crelle. Berlin.] 3, 160-187. https://gdz.sub.uni-goettingen.de/id/PPN243919689_0003?tify=%7B%22pages%22%3A%5B189%5D%2C%22view%22%3A%22info%22%7D
[2] Katsuya Miyake. COMPLEX MULTIPLICATION IN THE SENSE OF ABEL[C] NUMBER THEORY: Plowing and Starring Through High Wave Forms: Proceedings of the 7th China–Japan Seminar. 2015: 145-167.
[3] Umeno Ken. Theory of Complex Multiplication as a Cradle of Exactly Solvable Chaos[J]. Bulletin of the Japan Society for Industrial and Applied Mathematics (in Japanese), 2022, 32(4): 221-234. https://www.jstage.jst.go.jp/article/bjsiam/32/4/32_221/_pdf
[4] Katsuya Miyake. Origin of class field theory (number theory and its applications)[J]. RIMS Kôkyûroku, 1998, 1060: 185-209. https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1060-21.pdf
[5] Takase Masahito. Three Aspects of the Theory of Complex Multiplication[J]. The Intersection of History and Mathematics, 1994: 91-108.