Question #41611

Let $O$ be a fixed origin. A moving point $O_t$ has a position vector relative to $O$ given by: $\vec{r}_0(t) = \left( \int_{-1}^t \frac{u^2 - 1}{u^4 + 1} du + C \right) \hat{i} + t \hat{j} + t^2 \hat{k}$ The vertices of a tetrahedron $A, B, C, D$ have position vectors relative to $O_t$ such that their sum is: $\sum_{V \in \{A,B,C,D\}} \vec{r}_{V/O_t} = 4 \left( \int_t^1 \frac{u^2 - 1}{u^4 + 1} du \right) \hat{i} + 4(1-t) \hat{j} + 4(1-t^2) \hat{k}$ The parameter $t$ is restricted to the maximal domain $S$ where the function $f(t) = \int_0^t \frac{x^2 - 1}{x^4 + 1} dx$ is decreasing. The constant $C$ is chosen such that the maximum value of the $\hat{i}$-component of $\vec{r}_0(t)$ for $t \in S$ is $\sqrt{2} \ln(\sqrt{2} + 1)$. If the absolute position vector of the centroid $G$ of the tetrahedron is $\vec{r}_G = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$, find the value of $\alpha + \beta + \gamma$.