It's Time To Have Pi Today

Geometry Level 4

4(arctan(1x)+arctan(1x3))=π 4\left( \arctan\left( \dfrac { 1 }{ x } \right) + \arctan\left( \dfrac { 1 }{ { x }^{ 3 } } \right) \right) =\pi

Let xx be the positive real number satisfying the equation above and x3x{x}^{3}-x can be expressed as

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

where a,b,ca, b, c and dd are positive integers, with a,ba,b coprime and dd square-free.

Find the value of the expression below.

4(arctan(bc)+arctan(ad)) 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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