${\displaystyle \int \frac{dx}{x\left(x^7+1\right)}}$  is equal to

${\displaystyle \int \frac{dx}{x\left(x^7+1\right)}}$  which is of the form $\frac{1}{x}f\left(x^n\right)$

$={\displaystyle \int \frac{x^6}{x^7\left(x^7+1\right)}}dx,$  substitute $x^7=t$

$\Rightarrow 7x^{6}dx=dt$

$={\displaystyle \int \frac{\frac{1}{7}dt}{t\left(t+1\right)}}$

$=\frac{1}{7}{\displaystyle \int \left(\frac{1}{t}+\frac{-1}{t+1}\right)}dt$

$=\frac{1}{7}\log \left|\frac{t}{t+1}\right|+C$

$=\frac{1}{7}\log \left(\frac{x^7}{x^7+1}\right)+C$  assuming that$x>0$