Question : Is the following $(\star)$ true for $a,b,c\in\mathbb Z$ ?
$$\begin{align}\int_{0}^{\frac{\pi}{2}}(ax^4+b\pi x^3+c{\pi}^{2}x^2)\log(\sin x)dx=0\Rightarrow a=b=c=0\qquad(\star)\end{align}$$
Motivation : I've just been to able to prove the following theorem :
Theorem : If $(\star)$ is true, then $\frac{\zeta (5)}{\zeta (2)\zeta (3)}$ is an irrational number.
However, I can't prove that $(\star)$ is true. Can anyone help?
Remark : This question has been asked previously on math.SE without receiving any answers, where you can see the proof of the above theorem.