What is the proper "addition formula" for $ \int \frac{\mathrm{d}x}{\sqrt{1-x^6}} $?

Question

• The followings are two well-known "addition formulas" (or "addition theorems").

$$ \int_{0}^{u} \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^v \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_{0}^{u\sqrt{1-v^2}+v\sqrt{1-u^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}} $$ (which is just sine addition formula)

$$ \int_{0}^{u} \frac{\mathrm{d}x}{\sqrt{1-x^4}} + \int_0^v \frac{\mathrm{d}x}{\sqrt{1-x^4}} = \int_{0}^{\frac{u\sqrt{1-v^4}+v\sqrt{1-u^4}}{1+u^2 v^2}} \frac{\mathrm{d}x}{\sqrt{1-x^4}} $$ (by Euler)

• Both of the addition formulas are in the form of $$ \int_{0}^{u} \frac{\mathrm{d}x}{\sqrt{P(x)}} + \int_0^v \frac{\mathrm{d}x}{\sqrt{P(x)}} = \int_{0}^{some\ algebraic\ function\ of\ u\ and\ v} \frac{\mathrm{d}x}{\sqrt{P(x)}}$$ but no addition formula of this simple form exists for $ \int \frac{\mathrm{d}x}{\sqrt{1-x^6}} $ (essentially because $ y^2=1-x^6 $ is a genus $2$ curve). This was first shown by Abel in the 1820s.

• In fact, Abel proved the following theorem, which is stated slightly differently in various sources.

Skau, C. F., "Abelian integrals and the genesis of Abel's greatest discovery" (2020) (p.37)

Kleiman, S.L., "What is Abel's Theorem Anyway?", The Legacy of Niels Henrik Abel (2004) (p.417)

Gray, J. J., "Algebraic Geometry in the late Nineteenth Century", Proceedings of the Symposium on the History of Modern Mathematics, Vassar College, Poughkeepsie, New York, June 20–24, 1989 (p.366)

Cooke, R., "Abel's Theorem", Proceedings of the Symposium on the History of Modern Mathematics, Vassar College, Poughkeepsie, New York, June 20–24, 1989 (p.400)

• Most textbooks, if they ever do, mention this as a brief historical note and then quickly turns to the abstract language of Abel-Jacobi maps without giving any explicit examples of this theorem. I want to know how to constructively apply this theorem to the genus $>1$ case.

• Specifically, I want to know how this theorem applies to $ \int \frac{\mathrm{d}x}{\sqrt{1-x^6}} $, that is, what does the constructive result of Abel's addition theorem explicitly look like, after all the required computations? (I'm totally okay with using non-elementary functions, for instance some multivariable versions of theta or $ \wp $ functions if needed in the process; I just want to know the final result.)

Answer 1

Using \begin{align}\tag{1}\label{eq:1} I(u)=\int_0^u\frac {\mathrm d x}{\sqrt{1-x^6}}= u\,{}_2F_1\big(\tfrac16,\tfrac12;\tfrac76;u^6\big)\,, \end{align} we can tabulate the high-precision numerical solutions to $I(w)=I(u)+I(v)$ for $0<u,v<1$ and determine, for each $u,v$, the minimal polynomial for $w$ using Mathematica. The resulting polynomials with integer coefficients are quadratic in $w^2$ and can be reverse engineered (using FindSequenceFunction[] and a bit intuition) to reduce to the unique symmetric polynomial \begin{align}\tag{2}\label{eq:2} u^4+v^4+w^4 +2 \left(u^2 v^2+u^2 w^2+v^2 w^2\right) \left(2 u^2 v^2 w^2-1\right)=0 \,, \end{align} with solution \begin{align}\tag{3}\label{eq:3} w=\pm\sqrt{ \frac{u^2+v^2-2 u v \left[u^3v^3-\sqrt{\left(1-u^{6}\right)\left(1-v^{6}\right)}\right]} {1+4 u^2 v^2 \left(u^2+v^2\right)} } \,. \end{align} The positive solution is correct for $u+v\lesssim1$ (I did not determine the exact condition), while the negative solution hold for $u+v\gtrsim1$ provided that the constant $I_c=4^{1/3}\omega_2=4^{1/3}\Gamma(1/3)^3/(4\pi)$ is added to $I(w)$, where $\omega_2$ denotes a Weierstrass half period.

Finally some Mathematics code:

II[u_] = u Hypergeometric2F1[1/6, 1/2, 7/6, u^6];
ww[u_,v_] := ww[u,v] = w /. FindRoot[II[u]+II[v] == II[w],{w,.99},
    MaxIterations->500,WorkingPrecision->200]
tab = Simplify[Table[{u, v, MinimalPolynomial[RootApproximant[ww[u,v],4],w]},
    {u, 3/29, 16/29, 3/29}, {v, 3/31, 16/31, 1/31}]];
poly = FullSimplify@Table[
    {tab[[k,1,1]],FindSequenceFunction[tab[[k,All,{2,3}]],v]},{k,1,4}]
Simplify[poly==Table[Block[{u=tab[[k,1,1]]},{u,31^4 29^4
    (u^4+v^4+w^4 + 2(2u^2v^2w^2 - 1)(u^2v^2 + u^2w^2 + v^2w^2))}],{k,1,4}]]
$MaxExtraPrecision = 100;
Quiet@Chop[Table[Block[
    {w = +Sqrt[(u^2 + v^2 - 2 u v (u^3 v^3 - Sqrt[(1 - u^6) (1 - v^6)]))/
               (1 + 4 u^2 v^2 (u^2 + v^2))]},
    If[u+v<1, N[II[u]+II[v]-II[w] - 0 4^(1/3) Gamma[1/3]^3/(4 Pi), 20],""]],
    {u, 1/29, 28/29, 1/29}, {v, 3/31, 30/31, 1/31}],150]