An odd integral

Calculus Level 5

The following image shows the graph for \(f\left( x \right) =\dfrac { { x }^{ 2 } }{ { x }^{ 4 }+4 }\).

By drawing a tangent line at either absolute maximum for \(f\left( x \right)\), one can bound a finite region above \(f\left( x \right)\) that ends at the \(y\)-value for these maxima.

If the area of this region can be expressed in the form of \[\frac { 1 }{ a } \tanh ^{ -1 } \left ( { \frac { b\sqrt { c } }{ d } } \right) -\sin ^{ -1 } \left({ \frac { 1 }{ \sqrt { e } } } \right) +\frac { 1 }{ \sqrt { f } }\]

where \(a,b,c,d,e\) and \(f\) are positive integers with \(b,d\) coprime and \(c,e,f\) square-free, find \(a+b+c+d+e+f\).


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