Compute $\sum_{k \in \mathbb{Z}}\left|\int_{0}^{1} x^{2} e^{i k x} d x\right|^{2}$

Question

Compute $\sum_{k \in \mathbb{Z}}\left|\int_{0}^{1} x^{2} e^{i k x} d x\right|^{2}$

Do you have better solution for this problem without explicit compute this integral?

$\frac{e^{i k} ((1+i)-i k \log (e)) (k \log (e)+(1+i))-2 i}{k^3 \log ^3(e)} $