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 $a+b+c+d$ .