Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.

\(\mathbf{F}=abla f\), where \(f(x, y, z)=4 x^{2} \ln \left(1+y^{4}+z^{2}ight), \quad\) the path \(\mathbf{r}(t)=\left\langle t^{3}, \ln \left(1+t^{2}ight), e^{t}ightangle\) for \(0 \leq t \leq 1\)