$m{x}^2-2mx+m-1=0$
$\left(1\right)m\ne 0$
$\left(2\right)\Delta >0\to m\in \left(0,\infty \right)$
$\left(3\right)x_1\ne 0\wedge x_2\ne 0\to m\ne 1$

$\left(4\right)\frac{1}{x_1^3}+\frac{1}{x_2^3}>0$
        $\frac{\left(x_1+x_2\right)\left((x_1+x_2{)}^2-3x_1{x}_2\right)}{(x_1{x}_2{)}^3}>0$
        $\frac{\frac{2m}{m}\cdot \left({\left(\frac{2m}{m}\right)}^2-3\cdot \frac{m-1}{m}\right)}{{\left(\frac{m-1}{m}\right)}^3}>0$
        $\frac{2\cdot \left(4-3\cdot \frac{m-1}{m}\right)}{\frac{{\left(m-1\right)}^3}{m^3}}>0$
        $\left(8-\frac{6\left(m-1\right)}{m}\right)\cdot \frac{m^3}{{\left(m-1\right)}^3}>0$
        $\left(8-\frac{6\left(m-1\right)}{m}\right)\cdot \frac{m}{m-1}\cdot \frac{m^2}{{\left(m-1\right)}^2}>0$
        $\left(8-\frac{6\left(m-1\right)}{m}\right)\cdot \frac{m}{m-1}>0$
        $\frac{8m}{m-1}-6>0$

$\frac{8m}{m-1}-\frac{6\left(m-1\right)}{m-1}>0$
$\frac{8m-6m+6}{m-1}>0$
$\frac{2m+6}{m-1}>0$
$\left(2m+6\right)\left(m-1\right)>0$
$2m+6=0\vee m-1=0$
$m=-3\vee m=1$

$m\in \left(-\infty ,-3\right)\cup \left(1,\infty \right)$

Odp.: $m\in \left(1,\infty \right)$