${\left(x+\sqrt{x^3-1}\right)}^6+{\left(x-\sqrt{x^3-1}\right)}^6$

$=2\left({}^6{C}_0 x^6{ +}^{ 6}{C}_2{x}^4\left(x^3-1\right){+ }^6{C}_4 x^2{\left(x^3-1\right)}^2{ + }^6{C}_6{ \left(x^3-1\right)}^3\right)$

$=2\left({}^6{C}_0 x^6{ +}^{ 6}{C}_2 \left(x^7-x^4\right){+ }^6{C}_4 x^2\left(x^6-2x^3+1\right) {+}^6{C}_6 \left(x^9-3x^6+3x^3-1\right)\right)$

$\therefore$ Sum of the coefficient of even power of $x$

$=2\left({}^6{C}_0{-}^6{C}_2{+}^6{C}_4{+}^6{C}_4-3{-}^6{C}_6\right)$

$=2\left(1-15+15+15-3-1\right)=24$