Inspired by Hummus

Calculus

$\int _{ 0 }^{ 1 }{ \left\{ \dfrac { 1 }{ { x }^{1/6 } } \right\} \,dx } =\dfrac { A }{ B } -\dfrac { { \pi }^{ C } }{ D }$

The equation above holds true for positive integers $A,B,C,$ and $D$ , with $A$ and $B$ coprime.

Find $A+B+C+D$ .

Notation: $\{ \cdot \}$ denotes the fractional part function.