NOTE: I post this question on math.stackexchange but nobody answered, so I try here.

For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-_{3}F_{2}\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{5}{4},\frac{5}{4};-1\right).\tag{1}$$ Classical approaches seem to lead nowhere, but it is possible to translating the problem into the language of elliptic functions. Let $\text{sn}(u,k)$ be the Jacobi elliptic sine. We can prove that the evaluation of $(1)$ boils down to the evaluation of $$\int_{0}^{T/4}\log\left(-e^{-\pi i/4}\text{sn}\left(e^{3\pi i/4}z,-1\right)\right)dz$$ where $T=2K(1/2)$ and $K\equiv K(k)$ is the complete elliptic integral of the first kind with $k$ the elliptic modulus. I am not an expert in elliptic functions so I have difficulty to understand if this integral can be evaluated or not. However, I found this formula $$\log\left(\text{sn}\left(u,k\right)\right)=\log\left(\frac{2K}{\pi}\right)+\log\left(\sin\left(\frac{\pi u}{2K}\right)\right)-4\sum_{n\geq1}\frac{1}{n}\frac{q^{n}}{1+q^{n}}\sin^{2}\left(\frac{n\pi u}{2K}\right)$$ with $$q\equiv e^{-\pi\frac{K}{K^{\prime}}}=e^{\pi i\tau}$$ and $\left|\text{Im}\left(\frac{\pi u}{2K}\right)\right|<\frac{\pi}{2}\text{Im}\left(\tau\right)$. So, assuming that we can exchange the integral with the series, which I'm not sure about, the problem boils down to studying the following Lambert series $$\sum_{n\geq1}\frac{1}{n^{2}}\frac{q^{n}}{1+q^{n}}\sin\left(\frac{\pi nT}{4K}\right).\tag{2}$$ I have seen that similar series have been studied but this particular one has not (as far as I know). Clearly, there are a lot of heuristic passages and so I may have written nonsense.

**Questions**:

$1)$ Is it possible to find a closed form (in terms of special functions) of $(1)$?

$2)$ Assuming that the “elliptic approach” is correct, is there a closed form of $(2)$, maybe in terms of elliptic functions?

Thank you

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