\[ \large \displaystyle \int_{0}^{{\pi} / {4}}\cos^{10}x\sin{20x} \, dx \]

If the integral above is of the form \( \dfrac A B\), where \(A\) and \(B\) are coprime positive integers, find \(A+B\).

**Note**: Mathematical as well as Programming solution is acceptable.

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