ARML-Inspired Equation

Algebra Level 5

Suppose that xx is a real number such that 4tan1x+6tan13x=π.\begin{aligned} 4\tan^{-1} x+6\tan^{-1} 3x = \pi. \end{aligned} If x2=abcdx^2 = \dfrac{a-b\sqrt{c}}{d}, where a,b,c,da,b,c,d are positive integers with gcd(a,d)=gcd(b,d)=1\gcd(a,d)=\gcd(b,d)=1 and cc is not divisible by the square of any prime, find a+b+c+da+b+c+d.

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