Integral Integral11

Hint:

If $y \geq 1$ ,

$\int_y^{\infty } \left(\frac{1}{\left(\int_0^{\infty } \log \left(\frac{1}{x^k}+1\right) \, dx\right){}^2}-\frac{\sin \left(\frac{2 \pi }{k}\right)}{\pi k}\right) \, dk=-\frac{y \sin ^2\left(\frac{\pi }{y}\right)}{\pi ^2}$

using that

$\int_0^{\infty } \log \left(\frac{1}{x^k}+1\right) \, dx=\pi \csc \left(\frac{\pi }{k}\right)$

for $k>1$ .

Note that $\sin ^2\left(\frac{\pi }{y}\right)$ is easy to write in radicals at $y=64$ , since 64 is a power of 2.