I tried to simply find the torsion first by computing $- \frac{{\mathbf N\times B'}}{||\mathbf{r'(t)}||} $ and I was hoping to obtain a constant, but the algebra was too messy and so gave up on that method. I also tried to parametrise the curve with its natural parametrise but I couldn't find this due to t>0.
So I was wondering how this can be done.
I'll write $\;r(t)=\left(\,1+t\log t,\,-t\log t,\,t^2\,\right) , \;t>0\;$ , so the torsion at any point is given by
$$\tau(t)=\frac{\left(r'\times r''\right)\cdot r'''}{\left\|r'\times r''\right\|^2}$$
and
$$\begin{cases}&r'=(\,\log t+1,\,-\log t-1,\,2t\,)\\{}\\ &r''=\left(\,\frac1t,\,-\frac1t,\,2\,\right)\\{}\\ &r'''=\left(\,-\frac1{t^2},\,\frac1{t^2},\,0\,\right)\end{cases}\;\;\;\;\implies $$
$$\implies r'\times r''=\begin{vmatrix}i&j&k\\ 1+\log t&-1-\log t&2t\\ \frac1t&-\frac1t&2\end{vmatrix}=\left(\,-2\log t,\,-2\log t,\,0\,\right)$$
and
$$ \tau(t)=\frac1{8\log^2t}\left(\,\frac{2\log t}{t^2}-\frac{2\log t}{t^2}\,\right)=0$$