Are they nested?

Calculus Level 5

\[\int _{ 0 }^{ \infty }{ \left( \frac { { x }^{ 3 } }{ { e }^{ x }-\left( \frac { \sum _{ a=1 }^{ \infty }{ \left( \frac { 1 }{ a+12{ a }^{ 2 } } \right) } }{ \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2tan\left( \alpha \right) -2tan\left( \alpha \right) { \left( sin\left( \alpha \right) \right) }^{ \frac { 1 }{ 6 } } \right) } d\alpha } \right) } \right) } dx=\frac { 1 }{ A{ \pi }^{ B } } \]

Given that \(A\) and \(B\) are integers, find the value of \(A+B\).

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