Integrate $\int \frac{dx}{(x-1)\sqrt{x^2-x+1}}$ using any method

Question

I have an expression

$$\int \frac{1}{(x-1)\sqrt{x^2-x+1}}dx$$

to integrate, which I don't know how to integrate. I tried to factorize and solve, but got really ugly expression at the end.

Would appreciate any help.

Answer 1

That edit makes it easier. Let $x-\frac12=\frac{\sqrt3}2\tan\theta$. Then $$\begin{align}\int\frac{dx}{(x-1)\sqrt{x^2-x+1}} & =\int\frac{d\theta}{\frac{\sqrt3}2\sin\theta-\frac12\cos\theta}=\int\frac{d\theta}{\sin(\theta-\pi/6)} \\ & =-\ln(\csc(\theta-\pi/6)+\cot(\theta-\pi/6))+C_1 \\ & =\ln\left(\frac{\sin(\theta-\pi/6)}{1+\cos(\theta-\pi/6)}\right)+C_1 \\ & = \ln\left(\frac{\frac{\sqrt3}2\sin\theta-\frac12\cos\theta}{1+\frac{\sqrt3}2\cos\theta+\frac12\sin\theta}\right)+C_1\end{align}$$ From the Pythagorean theorem, $$\sin\theta=\frac{x-\frac12}{\sqrt{x^2-x+1}},\,\,\cos\theta=\frac{\frac{\sqrt3}2}{\sqrt{x^2-x+1}}$$ $$\begin{align}\int\frac{dx}{(x-1)\sqrt{x^2-x+1}} & =\ln\left(\frac{\frac{\sqrt3}2(x-\frac12)-\frac12\frac{\sqrt3}2}{\sqrt{x^2-x+1}+\frac{\sqrt3}2\frac{\sqrt3}2+\frac12(x-\frac12)}\right)+C_1 \\ & =\ln\left(\frac{x-1}{\sqrt{x^2-x+1}+\frac12x+\frac12}\right)+C\end{align}$$