\displaystyle{\int_0^{1/\sqrt{2}} \frac{x^{k-1}}{1-x^8}\,dx   =\int_0^{1/\sqrt{2}} \sum_{j=1}^\infty x^{8j+k-1}\,dx  = \frac{1}{2^{k/2}}\sum_{j=0}^\infty \frac{1}{16^j (8j+k)},}