Answer
$$x + \ln \left| {{x^3} + 1} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{x^3} + 3{x^2} + 1}}{{{x^3} + 1}}} dx \cr
& {\text{use long division}} \cr
& = \int {\left( {1 + \frac{{3{x^2}}}{{{x^3} + 1}}} \right)} dx \cr
& {\text{sum rule}} \cr
& = \int {dx} + \int {\frac{{3{x^2}}}{{{x^3} + 1}}} dx \cr
& {\text{find the antiderivatives}} \cr
& = x + \ln \left| {{x^3} + 1} \right| + C \cr} $$