Filled glasses

$f\left( x \right) =\sec ^{ 2 }{ x-\csc ^{ 2 }{ x } }$

The yellow region shown in the picture can be formed by bounding $f\left( x \right)$ with two horizontal lines at $y=8\sqrt { 3 }$ and $y=-8\sqrt { 3 }$ .

Considering the periodic properties of $f\left( x \right)$ , the area of any region like this one can be expressed in the form $\frac { a\pi }{ \sqrt { b } }$ , with $a$ a highly composite number and $b$ a prime integer. Find $\frac { a }{ b }$ .

This problem is original. The picture of the graph was produced from Desmos .