I am trying to calculate the potential for a system of two non-linear coupled oscillators. To do that I would have to calculate the following integral: \begin{equation} V(x,y)=\alpha \int_{}|x|^{n-1}x dx+ \beta \int_{}|y-x|^{n-1}(y-x)dx, \quad \alpha,\beta \in \mathbb{R} \end{equation} where $n \in [0,1]$.
Problem is I cannot really think of a way to analytically go throught with this integration. In fact, I am not even sure that there exists an analytical solution. Any opinions?