By partial fraction decomposition:

$$\int\frac{x}{(x^2+1)(x-1)}dx=-\int\frac{dx}{x}+2\int\frac{dx}{(x-1)^3}+\int\frac{dx}{(x-1)^2}+2\int\frac{dx}{x-1}=\boxed{\ln\frac{(x-1)^2}{|x|}}-\frac{x}{(x-1)^2}+C$$