Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals

Question

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Define the function $\mathcal{I}:\left(0,1\right)\rightarrow\mathbb{R}$ via the definite integral

$$\mathcal{I}{\left(p\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{4p}{\sqrt{\left(1-x^{3}\right)\left(1-p^{6}x^{3}\right)}}.\tag{1}$$

Question: Given $p\in\left(0,1\right)$, is there a (relatively) simple way to express $\mathcal{I}{\left(p\right)}$ in terms of elliptic integrals in Legendre normal form? Can $\mathcal{I}{\left(p\right)}$ possibly be written as a simple scalar multiple of a complete elliptic integral of the first kind?

Recall that for amplitude $\theta\in\left(0,\frac{\pi}{2}\right)$ and elliptic modulus $\kappa\in\left(0,1\right)$, the incomplete elliptic integral of the first kind $F{\left(\theta,\kappa\right)}$ is defined as

$$F{\left(\theta,\kappa\right)}:=\int_{0}^{\theta}\mathrm{d}\varphi\,\frac{1}{\sqrt{1-\kappa^{2}\sin^{2}{\left(\varphi\right)}}}.\tag{2}$$

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Motivation:

I have noticed that of the questions on this site related to proving special closed form values of the Gauss hypergeometric function, ${_2F_1}{\left(a,b;c;z\right)}$, an unusually large number of the more difficult ones can be shown equivalent to proving the particular hypergeometric function with parameters $a=\frac12$, $b=\frac13$, and $c=\frac56$, i.e., ${_2F_1}{\left(\frac12,\frac13;\frac56;z\right)}$, takes some special (usually algebraic) value for some particular (usually rational) value of $z$, where $0<z<1$. So in the interest of generalization, I wanted to see how close one could get to obtaining a closed form for $f(z):={_2F_1}{\left(\frac12,\frac13;\frac56;z\right)}$ for arbitrary values of $z\in(0,1)$.

Using Euler's integral representation for the Gauss hypergeometric function with the assumption that $0<z<1$, we have:

$$\begin{align} f(z) &={_2F_1}{\left(\frac12,\frac13;\frac56;z\right)}\\ &=\frac{1}{\operatorname{B}{\left(\frac13,\frac12\right)}}\int_{0}^{1}t^{-\frac23}(1-t)^{-\frac12}(1-zt)^{-\frac12}\mathrm{d}t\\ &=\frac{3}{\operatorname{B}{\left(\frac13,\frac12\right)}}\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{(1-x^3)(1-zx^3)}},\tag{3}\\ \end{align}$$

where the beta function factor can alternatively be written in terms of the gamma function as,

$$\operatorname{B}{\left(\frac13,\frac12\right)}=\frac{\sqrt{3}}{2^{4/3}\pi}\left[\Gamma{\left(\frac13\right)}\right]^3.$$

The integral factor in the last line of $(3)$ is already superficially quite similar to the following integral representation of the complete elliptic integral of the first kind:

$$K(k)=\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$

Moreover, the last line of $(3)$ implies that in order for $f(z)$ to take an algebraic value, the integral factor will be the product of an algebraic number $A$ with an elliptic integral singular value:

$$\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{(1-x^3)(1-zx^3)}}=A\,K{\left(k_3\right)}=A\frac{3^{1/4}}{2^{7/3}\pi}\left[\Gamma{\left(\frac13\right)}\right]^3.$$

This made me suspect that the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{(1-x^3)(1-zx^3)}}=\mathcal{I}{(z^{1/6})}$ might be capable of being written as a linear combination of complete elliptic integrals in general.

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Answer 1

General Procedure:

I will give an outline of a possible procedure (that I learned from a calculus book some time ago) to solve integrals of the form $$\mathcal{J}=\int\frac{{\rm d}x}{\sqrt{(x^2+ax+b)(x^2+cx+d)}}$$ First, let $x=t+\lambda$. \begin{align} \mathcal{J}=\int\frac{{\rm d}t}{\sqrt{(t^2+(2\lambda+a)t+\lambda^2+\lambda a+b)(t^2+(2\lambda+c)t+\lambda^2+\lambda c+d)}} \end{align} Next, let $\lambda=\dfrac{d-b}{a-c}$, then make the substitution $\displaystyle t=\mu v$ where $\displaystyle \mu=\sqrt{\lambda^2+\frac{ad-bc}{a-c}}$. $$\mathcal{J}=\frac{1}{\mu}\int\frac{{\rm d}v}{\sqrt{\left(v^2+Av+1\right)\left(v^2+Cv+1\right)}}$$ where $\displaystyle A=\frac{2\lambda+a}{\mu}$ and $\displaystyle C=\frac{2\lambda+c}{\mu}$. Follow this up with the substitution $\displaystyle v=\frac{1+w}{1-w}$. $$\mathcal{J}=\frac{2}{\mu}\int\frac{{\rm d}w}{\sqrt{((2-A)w^2+(2+A))((2-C)w^2+(2+C))}}$$ By applying the appropriate trigonometric substitution, it can be seen that $\mathcal{J}$ can be reduced to incomplete elliptic integrals.

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Parameters

I will define the following parameters to save space and typing. \begin{align} \alpha_1=&\ \frac{1}{\sqrt{\left(a+1\right)^2\left(3a^2+2\sqrt{3}\sqrt{a^4+a^2+1}+3\right)}}\\ \alpha_2=&\ \frac{1}{\sqrt{\left(a-1\right)^2\left(3a^2+2\sqrt{3}\sqrt{a^4+a^2+1}+3\right)}}\\ \beta_1=&\ \frac{\sqrt{3}a^2-\sqrt{a^4+a^2+1}+\sqrt{3}a+\sqrt{3}}{\sqrt{3}a^2+\sqrt{a^4+a^2+1}+\sqrt{3}a+\sqrt{3}}\\ \beta_2=&\ \frac{\sqrt{3}a^2-\sqrt{a^4+a^2+1}-\sqrt{3}a+\sqrt{3}}{\sqrt{3}a^2+\sqrt{a^4+a^2+1}-\sqrt{3}a+\sqrt{3}}\\ \xi_1=&\ \frac{\sqrt{a^4+a^2+1}-\sqrt{3}a}{\sqrt{a^4+a^2+1}+\sqrt{3}a}\\ \xi_2=&\ \frac{\sqrt{a^4+a^2+1}+\sqrt{3}a}{\sqrt{a^4+a^2+1}-\sqrt{3}a} \end{align}

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Evaluation of $\ \mathcal{I}(a)$

It was derived by the OP that \begin{align} \mathcal{I}(a) =&\underbrace{\frac{1}{4a}\int^\infty_\frac{a^2+1}{2a}\frac{{\rm d}x}{\sqrt{\left(x^2-\frac{(a-1)^2}{2a}x-\frac{a^2+1}{2a}\right)\left(x^2+\frac{a^2+1}{2a}x+\frac{a^4-a^2+1}{4a^2}\right)}}}_{\mathcal{I}_1}\\ &+\underbrace{\frac{1}{4a}\int^\infty_\frac{a^2+1}{2a}\frac{{\rm d}x}{\sqrt{\left(x^2-\frac{(a+1)^2}{2a}x+\frac{a^2+1}{2a}\right)\left(x^2+\frac{a^2+1}{2a}x+\frac{a^4-a^2+1}{4a^2}\right)}}}_{\mathcal{I}_2} \end{align} Applying the aforementioned procedure and using Wolfram Alpha to do all the algebra, we get \begin{align} \mathcal{I}_1 =&\frac{1}{4a}\int^\infty_\frac{a^2+1}{2a}\frac{{\rm d}x}{\sqrt{\left(x^2-\frac{(a-1)^2}{2a}x-\frac{a^2+1}{2a}\right)\left(x^2+\frac{a^2+1}{2a}x+\frac{a^4-a^2+1}{4a^2}\right)}}\\ =&\frac{1}{\sqrt{3}\sqrt{a^4+a^2+1}}\int^\infty_\frac{\sqrt{3}(a^2+a+1)}{\sqrt{a^4+a^2+1}}\frac{{\rm d}x}{\sqrt{\left(x^2-\frac{2(2a^2+a+2)}{\sqrt{3}\sqrt{a^4+a^2+1}}x+1\right)\left(x^2-\frac{2\sqrt{3}\ a}{\sqrt{a^4+a^2+1}}+1\right)}}\\ =&\alpha_1\int^1_{\beta_1}\frac{{\rm d}x}{\sqrt{\left(x^2-\beta_1^2\right)\left(x^2+\xi_1\right)}} =\alpha_1\int^{\arccos{\beta_1}}_{0}\frac{\sec{\varphi}\tan{\varphi}}{\sqrt{(\sec^2{\varphi}-1)(\beta_1^2\sec^2{\varphi}+\xi_1)}}{\rm d}\varphi\\ =&\alpha_1\int^{\arccos{\beta_1}}_{0}\frac{{\rm d}\varphi}{\sqrt{\beta_1^2+\xi_1\cos^2{\varphi}}} =\alpha_1\int^{\arccos{\beta_1}}_{0}\frac{{\rm d}\varphi}{\sqrt{(\beta_1^2+\xi_1)-\xi_1\sin^2{\varphi}}}\\ =&\frac{\alpha_1}{\sqrt{\beta_1^2+\xi_1}}\int^{\arccos{\beta_1}}_{0}\frac{{\rm d}\varphi}{\sqrt{1-\frac{\xi_1}{\beta_1^2+\xi_1}\sin^2{\varphi}}} =\color{red}{\frac{\alpha_1}{\sqrt{\beta_1^2+\xi_1}}\mathbf{F}\left(\arccos{\beta_1},\sqrt{\frac{\xi_1}{\beta_1^2+\xi_1}}\right)} \end{align} for $\mathcal{I}_1$, and we obtain (I omit the steps of trigonometric manipulations since they are the same) \begin{align} \mathcal{I}_2 =&\frac{1}{4a}\int^\infty_\frac{a^2+1}{2a}\frac{{\rm d}x}{\sqrt{\left(x^2-\frac{(a+1)^2}{2a}x+\frac{a^2+1}{2a}\right)\left(x^2+\frac{a^2+1}{2a}x+\frac{a^4-a^2+1}{4a^2}\right)}}\\ =&\frac{1}{\sqrt{3}\sqrt{a^4+a^2+1}}\int^\infty_{\frac{\sqrt{3}(a^2-a+1)}{\sqrt{a^4+a^2+1}}}\frac{{\rm d}x}{\sqrt{\left(x^2-\frac{2(2a^2-a+2)}{\sqrt{3}\sqrt{a^4+a^2+1}}x+1\right)\left(x^2+\frac{2\sqrt{3}\ a}{\sqrt{a^4+a^2+1}}x+1\right)}}\\ =&\alpha_2\int^1_{\beta_2}\frac{{\rm d}x}{\sqrt{\left(x^2-\beta_2^2\right)\left(x^2+\xi_2\right)}}=\color{blue}{\frac{\alpha_2}{\sqrt{\beta_2^2+\xi_2}}\mathbf{F}\left(\arccos{\beta_2},\sqrt{\frac{\xi_2}{\beta_2^2+\xi_2}}\right)} \end{align} Therefore $$\mathcal{I}(a)=\frac{\alpha_1}{\sqrt{\beta_1^2+\xi_1}}\mathbf{F}\left(\arccos{\beta_1},\sqrt{\frac{\xi_1}{\beta_1^2+\xi_1}}\right)+\frac{\alpha_2}{\sqrt{\beta_2^2+\xi_2}}\mathbf{F}\left(\arccos{\beta_2},\sqrt{\frac{\xi_2}{\beta_2^2+\xi_2}}\right)$$ I have substituted in some random values of $a$ into $\mathcal{I}_1$ and it seems to match numerically. However, I have not verified $\mathcal{I}_2$ yet so there might be a mistake.