Integrating $\int \left(\sqrt[6]{\frac{x}{x-2}} - \sqrt[4]{\frac{x}{x-2}}\right)\frac{\mathrm dx}{x^2-2x}$ with partial fractions

Question

Recently I've been studying on partial fractions and integration using partial fraction decomposition. I've not had any problems solving those types of integrals until I came across this integral:

$$ \int \left(\sqrt[6]{\dfrac{x}{x-2}} - \sqrt[4]{\dfrac{x}{x-2}}\right)\frac{\mathrm dx}{x^2-2x}$$

The book hints that you should substitute $\left( \dfrac{x}{x-2}=t^{12}, t \in \Bbb R \right)$. I've tried countless times but haven't found any way as to even end up with an integrand that can be decomposed into partial fractions.

Answer 1

With you Substitution you will get $$x=\frac{-2t^{12}}{1-t^{12}}$$ and you can compute $$dx=…$$

Answer 2

Hint

For $\dfrac x{x-2}\ge0,$ we need either $x>2$ or $x<0$

So, WLOG $x-1=\sec2t,dx=?$

$\dfrac x{x-2}=\dfrac{1+\sec2t}{\sec2t-1}=\tan^2t$

$x^2-2x=\tan^22t$