To solve \(\int \left( x^3 + \frac{1}{x^3} \right) \, dx\), we integrate each term separately.

First, integrate \(x^3\):

\(\int x^3 \, dx = \frac{x^4}{4} + C_1\).

Next, integrate \(\frac{1}{x^3} = x^{-3}\):

\(\int x^{-3} \, dx = \frac{x^{-2}}{-2} + C_2 = -\frac{x^{-2}}{2} + C_2\).

Combine the results:

\(\int \left( x^3 + \frac{1}{x^3} \right) \, dx = \frac{x^4}{4} - \frac{x^{-2}}{2} + c\), where \(c = C_1 + C_2\).