Since ${\sin }^2(\pi (x-[x]))={\sin }^2(\pi x)$ for all $x$, the factor ${\log }_e(1+{\sin }^2(\pi x))$ is continuous on all of $\mathbb{R}$ and vanishes precisely at integer $x$. The factor $[x^3]$ jumps wherever $x^3$ is an integer.

Set A (discontinuities of $f$): For $x\in (-3,3)$, we have $x^3\in (-27,27)$. The integer values that $x^3$ attains are $-26,-25,\dots ,-1,0,1,\dots ,26$, giving 53 candidate jump points of $[x^3]$.

At an integer $x=n$, both factors of $f$ are well-behaved and ${\log }_e(1+{\sin }^2(\pi n))=0$, which smooths out the jump of $[x^3]$. So integer $x$ values in $(-3,3)$, namely $x\in \{-2,-1,0,1,2\}$ (5 values), are NOT discontinuities.

The remaining $53-5=48$ jump points of $[x^3]$ (where $x^3$ is a nonzero integer but $x$ itself is non-integer) make $f$ discontinuous. So $|A|=48$.

Set B (discontinuities of $g$): $g(x)=x^3{\sin }^2(\pi {\log }_e(1+\{x\}))$ where $\{x\}=x-[x]$. The fractional part $\{x\}$ jumps at integer $x$. The function $h(t)={\sin }^2(\pi {\log }_e(1+t))$ has $h(0)=0$ and $h(1)={\sin }^2(\pi {\log }_{e}2)\ne 0$.

At a non-zero integer $n$, the left limit involves $\{x\}\to 1$ giving $h\to {\sin }^2(\pi {\log }_{e}2)\ne 0$, while at $x=n$ exactly $\{x\}=0$ giving $h=0$. The factor $x^3=n^3\ne 0$ at non-zero integers, so $g$ is discontinuous there. At $x=0$, both $x^3=0$ and $h=0$, and one can check continuity holds. So $B=\{-2,-1,1,2\}$, giving $|B|=4$.

Intersection: $A$ contains non-integer points only, $B$ contains integer points only, so $A\cap B=\varnothing$, $|A\cap B|=0$.

$|A|+2|B|-|A\cap B|=48+8-0=56$.