Oh no!

Calculus Level 5

\[\int _{ -\pi }^{ \pi }{ \left( \arcsin { \sin { x } } \right) } \left( \arcsin { \cos { x } } \right) \left( \arccos { \cos { x } } \right) \left( \arccos { \sin { x } } \right) \, dx\]

If the integral above can be expressed in the form \(\dfrac { { \pi }^{ \alpha } }{ \beta }\), where \({ \alpha }\) and \({ \beta }\) are positive integers, find \(\dfrac { \beta }{ \alpha }\).


If you have not done it yet, try part 1 first.
This problem is original.
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