# 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 $$a+b+c+d+e+f$$.

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