It's Time To Have Pi Today

Geometry Level 4

\[ 4\left( \arctan\left( \dfrac { 1 }{ x } \right) + \arctan\left( \dfrac { 1 }{ { x }^{ 3 } } \right) \right) =\pi \]

Let \(x\) be the positive real number satisfying the equation above and \({x}^{3}-x\) can be expressed as

\[ \dfrac { a }{ b } \left( c+\sqrt { d } \right) \; , \]

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

Find the value of the expression below.

\[ 4\left( \arctan\left( \dfrac { b }{ c } \right) + \arctan\left( \dfrac { a }{ { d } } \right) \right) \]

Give your answer to 4 decimal places.

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