Proving a formula for $\int_0^\infty \frac{\log(1+x^{4n})}{1+x^2}dx $ if $n=1,2,3,\cdots$

I came across the formula

$$\int_0^\infty \frac{\log \left(1+x^{4n} \right)}{1+x^2}dx = \pi \log \left\{2^n \prod_{k=1 ,\ k \text{ odd}}^{2n-1} \left(1+\sin \left( \frac{\pi k}{4n}\right) \right)\right\} $$

where $n=1,2,3,4,\cdots$

I checked it for a few values of $n$ and it seems to give correct results.

Please help me prove this result.

Using parity, write the integral as $\frac12\int_{-\infty}^{\infty}$ and try to move the contour of integration to $i\infty$. The obstructions will be:

• A simple pole at $x=i$ which will produce the contribution $$2\pi i \,\mathrm{res}_{z=i}\frac{\ln(1+x^{4n})}{1+x^2}=\pi\ln 2.\tag{1}$$

• $2n$ branch cuts $B_k$ (with $k=0,\ldots,2n-1$) emanating from the logarithmic branch points $x_k=\exp\left\{ i\pi\frac{1+2k}{4n}\right\}$. These will produce integrals of the logarithm jumps (all equal to $2\pi i$) over the branch cuts. These integrals will have the simplest form if the branch cuts are given by radial rays $B_k=[x_k,x_k\infty)$. Then \begin{align}\int_{B_k}=2\pi i\int_{x_k}^{x_k\infty}\frac{dx}{1+x^2}&=\pi \left[\ln\frac{1+ix}{1-ix}\right]_{x=x_k}^{x=x_k\infty}=\pi\left[i\pi-\ln\frac{1+i x_k}{1-ix_k}\right]. \end{align}

The sum of branch cut integrals can be transformed to \begin{align} \sum_{k=0}^{2n-1}\int_{B_k}=\pi\left[2\pi i n-\sum_{k=0}^{2n-1}\ln\frac{1+i x_k}{1-ix_k}\right]&=\pi\sum_{k=0}^{2n-1}\ln\left|\frac{1-i x_k}{1+ix_k}\right|=\\ &=\frac{\pi}{2}\sum_{k=0}^{2n-1}\ln\frac{1+\sin(\arg x_k)}{1-\sin(\arg x_k)}=\\ &=\pi\sum_{k=0}^{2n-1}\ln\frac{1+\sin(\arg x_k)}{|\cos(\arg x_k)|}=\\ &=\pi\ln \left(2^{2n-1}\prod_{k=0}^{2n-1}(1+\sin(\arg x_k))\right).\tag{2} \end{align} At the last step of obtaining (2), we needed to calculate the product \begin{align} P=\prod_{k=0}^{2n-1}\cos^2(\arg x_k)=2^{-4n}\prod_{k=0}^{2n-1}(x_k-\bar{x}_k)^2= 2^{-4n}\prod_{k=0}^{2n-1}\left(e^{\frac {i\pi} {2 n}} + e^{-\frac {i\pi k} {n}}\right) \left(e^{-\frac {i\pi} {2 n}} + e^{\frac {i\pi k} {n}}\right). \end{align} But since $\displaystyle x^{2n}-1=\prod_{k=0}^{2n-1}\left(-x+e^{-\frac {i\pi k} {n}}\right)$, the products can be computed easily so that $$P=2^{-4n}\left(e^{i\pi}-1\right)\left(e^{-i\pi}-1\right)=2^{2-4n}.$$

Summing (1) and (2) and taking into account the factor of $\frac12$, we obtain the following expression for the integral: $$\int_0^{\infty}\frac{\ln(1+x^{4n})}{1+x^2}dx=\frac{\pi}{2}\ln \left(2^{2n}\prod_{k=0}^{2n-1}(1+\sin(\arg x_k))\right).$$ This is obviously equivalent to the quoted answer.