Let's integrate

$\int _{ 0 }^{ \infty }{ \left\lfloor \frac { 6 }{ { e }^{ x } } \right\rfloor \, dx } =\ln { \frac { { A }^{ B } }{ C! } }$

The above equation is true for positive integers $A$, $B$ and $C$.

Find the minimum value of $A+B+C$.

Bonus: Find the generalization of $\displaystyle \int _0^\infty \left \lfloor \frac n{e^x} \right \rfloor \, dx$.

Notation: $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$.