Greetings from a new-comer ...

I'm more a combinatorial geometer than an real analyst, so I'm seeking assistance with an integral-based conundrum.

Define these (special-to-me) functions for \(1/3 \leq x \leq 1/2\):

$$\begin{align}L_3(x) &:= \frac{3}{2} \int_{1/3}^x \; \frac{1}{\sqrt{t(1-t)}} \;\operatorname{atanh}\sqrt{\frac{3t-1}{1-t}} \; dt \\[8pt]

H_3(x) &:= 6 \int_{1/3}^x \;\; \frac{1}{\sqrt{1-t^2}} \;\; \operatorname{atanh}\sqrt{\frac{3t-1}{1-t}} \; dt

\end{align}$$

I'm looking for (non-trivial!) somehow-related functions \(f\) and \(g\) such that \(\displaystyle f(H_3) \equiv g(L_3)\) (that is, equal for all \(x\)).

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The context of this problem is hyperbolic 4-space, but let's start a little simpler.

In the hyperbolic plane, a right triangle with legs of length \(A_1\) and \(B_1\) and hypotenuse of length \(H_1\) gives rise to the hyperbolic Pythagorean Theorem

$$\begin{equation}\cosh H_1 = \cosh A_1 \cosh B_1 \end{equation}$$

In hyperbolic 3-space, a "right-corner" tetrahedron with right-triangle leg-faces of area \(A_2\), \(B_2\), \(C_2\) and hypotenuse-face of area \(H_2\) gives rise to this relation

$$\begin{equation}\cos\frac{H_2}{2} = \cos{\frac{A_2}{2}}\cos{\frac{B_2}{2}}\cos{\frac{C_2}{2}} - \sin{\frac{A_2}{2}}\sin{\frac{B_2}{2}}\sin{\frac{C_2}{2}} \end{equation}$$

In hyperbolic 4-space, the analogous simplex with right-corner leg-tetrahedra of volume \(A_3\), \(B_3\), \(C_3\), \(D_3\) and hypotenuse-tetrahedron of volume \(H_3\) gives rise to ... no known relation. Ultimately, I want to find the general relation here (if there is one), but for now, I'm concentrating on the special "isosceles" case, where \(A_3 = B_3 = C_3 = D_3 =: L_3 \). The integrals for \(L_3\) and \(H_3\) reduce to the formulas given above. (The parameter \(x\) is related to a dihedral angle in the configuration.)

By analogy with the lower-dimensional cases, when I say that I'm looking for \(f(H_3)\equiv g(L_3)\) for "somehow-related" functions \(f\) and \(g\), I'm anticipating \(f\) to be "cosine-like", and for \(g\) to effectively be a four-parameter function \(g(w, x, y, z)\) with all parameters equal (for now) and with "cosine-like" (and/or "sine-like") effects on each parameter. However, given the dramatic difference between the 2- and 3-dimensional identities, I really have no idea what I should be expecting here.

I'll close by mentioning the special-special case where \(x = 1/2\), for which we get

$$H_3^\star := \operatorname{Cl}_2\left(\frac{\pi}{3}\right) = \sum_{k=1}^{\infty} \frac{1}{k^2} \sin\frac{\pi k}{3} = 1.01494\dots \qquad

L_3^\star := \frac{1}{2} \operatorname{Cl}_2 \left(\frac{\pi}{2}\right) = \frac{1}{2}\; \sum_{k=1}^{\infty} \frac{1}{k^2} \sin\frac{\pi k}{2} = 0.45978\dots$$

where \(\operatorname{Cl}_2\) is one of the Clausen functions, coincidentally the subject of a thread by @DreamWeaver in the Tutorials section of this forum. If there's going to be an \(f\) and \(g\) that work at all, then they're going to have to work for \(H^\star_3\) and \(L^\star_3\). I've scanned DreamWeaver's tutorials (and other listings of relations among these kinds of functions), but I don't see an obvious connection here.

Any and all insights greatly appreciated.