ARML-Inspired Equation

Algebra Level 5

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