I'm curious if there's a faster way to calculate the Wronskian of the following five functions by hand: $$ y_1 = e^{r_1 x},\quad y_2 = e^{r_2 x},\quad y_3 = e^{r_3 x},\quad y_4 = e^{\alpha x}\cos(\beta x),\quad y_5 = e^{\alpha x}\sin(\beta x) $$ where $r_1, r_2, r_3$ are all distinct, and $\alpha\neq 0$. The result I'm expecting, thanks to Wolfram Alpha, is $$ -\beta(r_1 - r_2)(r_1 - r_3)(r_2 - r_3)((\alpha - r_1)^2 + \beta^2)((\alpha - r_2)^2 + \beta^2)((\alpha - r_3)^2 + \beta^2)e^{(2a+r_1+r_2+r_3)x} $$ the problem is that calculating this determinant by hand involves calculating (and then simplifying) $5! = 120$ terms, which seems unreasonable.
I recognize there's a lot of simplifications we can make to the derivatives of the $y_i$'s themselves. Obvious $y_i^{(k)} = r_i^ke^{r_i x}$ ($i = 1, 2, 3$), and the derivatives of $y_4$ and $y_5$ are related by $$ \left[\begin{matrix}y_4^{(k)} \\ y_5^{(k)}\end{matrix}\right] = \left[\begin{matrix} \alpha & -\beta \\ \beta & \alpha\end{matrix}\right]^k\left[\begin{matrix}y_4 \\ y_5\end{matrix}\right] $$ but I'm not sure how to utilize this information to simplify the calculation of the determinant. Can I use determinant identities in order to reduce this to, say, the determinant of an upper triangular matrix or something?
Any help would be appreciated!