Integral of irrational fraction to find right substitution

Question

I have to calculate the following integral by using a substitution which result in no so long calculations. What would it be?

$$\int \frac {\sqrt{x-1}-2} {((x-1)^{1/4}-1) \sqrt{x-1 }}\ \mathrm d x$$

Answer 1

HINT:

$\sqrt[4]{x-1}=t\implies\sqrt{x-1}=t^2,x-1=t^4\implies dx=4t^3\ dt$

$$\int\dfrac{t^2-2}{t^2(t-1)}4t^3dt=4\int\dfrac{t^3-2t}{t-1}dt$$

Now $t^3-2t=t^3-1-2(t-1)-1=(t-1)(t^2+t+1)-2(t-1)-1$

Answer 2

Hint: Substitute $$\sqrt{x-1}=t$$ then $$dx=2tdt$$ and our integral will be $$\int\frac{2t(t-2)}{t^{3/4}-t}dt$$ ,then substitute $$u=t^{1/4}$$,doing this we obtain $$\int \frac{8u^4(u^4-2)}{1-u}du$$