\[\large\mathfrak{T}=\displaystyle\int_0^{\pi/6}\left(\sum_{r=1}^{\infty}(-1)^{r+1}\cos x\sin^rx\right)\mathrm{d}x\]

If \(\large\mathfrak T\) can be expressed in the form \(\large\ln\left(\dfrac{\alpha e^{\frac{1}{\psi}}}{\gamma}\right)\), where \(\alpha,\psi,\gamma\) are primes, then:

\[\Large \alpha\times \psi\times \gamma=\ ?\]

**Details and Assumptions:-**

\(\large e\) is the Euler's number defined as: \[\displaystyle e = \lim_{n\to\infty} \left( 1 + \dfrac{1}{n} \right)^n \]

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