# Evaluating ∫10logxlog(1−x4)1+x2dx\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx

I am trying to prove that

$$∫10log(x)log(1−x4)1+x2dx=π316−3Glog(2)$$\int_{0}^{1}\frac{\log\left(x\right) \log\left(\,{1 - x^{4}}\,\right)}{1 + x^{2}} \,\mathrm{d}x = \frac{\pi^{3}}{16} - 3\mathrm{G}\log\left(2\right) \tag{1}$$$$

where $$G\mathrm{G}$$ is Catalan’s Constant.

I was able to express it in terms of Euler Sums but it does not seem to be of any use.

∫10log(x)log(1−x4)1+x2dx= 116∞∑n=1ψ1(1/4+n)−ψ1(3/4+n)n\begin{align} &\int_{0}^{1}\frac{\log\left(x\right) \log\left(\,{1 - x^{4}}\,\right)}{1 + x^{2}} \,\mathrm{d}x \\[3mm] = &\ \frac{1}{16}\sum_{n = 1}^{\infty} \frac{\psi_{1}\left(1/4 + n\right) - \psi_{1}\left(3/4 + n\right)}{n} \tag{2} \end{align}

Here $$ψn(z)\psi_{n}\left(z\right)$$ denotes the polygamma function.

Can you help me solve this problem $$??$$.

where $H_k$ are the Harmonic numbers. Then