To 306 followers! (Corrected)

Calculus Level 5

\[\int _{ 1 }^{ \infty }{ \frac { 1 }{ x^{ 2 } } \left( \frac {( \frac { d }{ dx } \cos ^{ -1 } \left( \cos \left( x\pi \right) \right)) +\pi}{ 2 \pi } \right) } dx=A \]

Given the above equation, find \(\left\lfloor 1000A \right\rfloor \).

Note: Sorry to those few who attempted the previous problem that I posted. The answer was wrong so I have posted a new problem.
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