Example 1 Solve the inequality $\log _{8}\left(x^{3}+x+3\right)>\log _{2}(x+1)$.
Solving the original inequality can be transformed into:
$$
\left\{\begin{array}{l}
x^{3}+x+3>0, \\
x+1>0, \\
x^{3}+x+3>(x+1)^{3},
\end{array}\right.
$$
which is equivalent to
$$
\begin{array}{l}
\left\{\begin{array}{l}
x+1>0, \\
3 x^{2}+2 x-2-1, \\
\frac{-1-\sqrt{7}}{3}<x<\frac{-1+\sqrt{7}}{3},
\end{array}\right. \\
\end{array}
$$

Solving this, we get $-1<x<\frac{-1+\sqrt{7}}{3}$.
Therefore, the solution set of the original inequality is: $\left\{x \left\lvert\,-1<x<\frac{-1+\sqrt{7}}{3}\right.\right\}$.