Question #72351

Let $\vec{r}(t)$ be a twice differentiable vector-valued function such that $\vec{r} \times \vec{r}' \neq \vec{0}$. Evaluate the indefinite integral: $\int \frac{ \left( |\vec{r}|^4 |\vec{r}'|^2 - |\vec{r}|^2 (\vec{r} \cdot \vec{r}')^2 \right) (\vec{r} \times \vec{r}'') - 2(\vec{r} \cdot \vec{r}') |\vec{r} \times \vec{r}'|^2 (\vec{r} \times \vec{r}') }{ |\vec{r}|^6 |\vec{r}'|^2 - |\vec{r}|^4 (\vec{r} \cdot \vec{r}')^2 } \, dt$ (where $C$ is an arbitrary scalar constant and $\vec{C}$ is an arbitrary constant vector)