I am having trouble finding a resource online that derives the expected Fisher's Information Matrix for the uni-variate Student's t-distribution. Does anyone know of such a resource?

In absence of any existing resource that derives the expected Fisher's information matrix for the t-distribution, I am trying to derive it myself but I'm stuck. Here is my work so far:

$y\_i \sim t(\mu, \sigma^2, v)$ where $v$ is the degrees of freedom (df) parameter (assumed fixed). Then:
\begin{align\*}
f(y\_i) &= \frac{\Gamma(\frac{v+1}{2})}{\Gamma(\frac{v}{2})\sqrt{\pi v \sigma^2}}\big(1+\frac{1}{v\sigma^2}(y\_i-\mu)^2\big)^{\frac{-(v+1)}{2}} 
\end{align\*}

Thus we have the following **log-likelihood function**:
\begin{align\*}
log f(y\_i)=log\Gamma(\frac{v+1}{2})-log\Gamma(\frac{v}{2})-\frac{1}{2}log(\pi v \sigma^2)+ \frac{-(v+1)}{2}log\big[1+\frac{1}{v\sigma^2}(y\_i-\mu)^2\big]
\end{align\*}

Here the **first derivative equations**:

\begin{align\*}
&\frac{\partial}{\partial \mu}logf(y\_i)=\frac{v+1}{2}\frac{\frac{2}{v\sigma^2}(y\_i-\mu)}{1+\frac{1}{v\sigma^2}(y\_i-\mu)^2} \\ & \frac{\partial}{\partial \sigma^2}logf(y\_i)= \frac{-1}{2\sigma^2}-\frac{(v+1)}{2} \frac{\frac{-1}{v\sigma^4}(y\_i-\mu)^2}{1+\frac{1}{v\sigma^2}(y\_i-\mu)^2}
\end{align\*}

And here are the **2nd derivative equations:**

\begin{align\*}
&\frac{\partial}{\partial \mu^2}logf(y\_i)=\frac{v+1}{2}\frac{\frac{-2}{v\sigma^2}+\frac{2}{dv^2\sigma^4}(y\_i-\mu)^2}{\big(1+\frac{1}{v\sigma^2}(y\_i-\mu)^2\big)^2} \\ & \frac{\partial}{\partial \mu \partial \sigma^2}logf(y\_i) =\frac{v+1}{2} \Big\{[\frac{2}{v\sigma^2}-\frac{4}{v^2\sigma^6}(y\_i-\mu)^2][1+\frac{1}{v\sigma^2}(y\_i-\mu)^2]^2-[\frac{-2}{v\sigma^2}+\frac{2}{v^2\sigma^4}(y\_i-\mu)^2]\*2[1+\frac{1}{v\sigma^2}(y\_i-\mu)^2][\frac{-1}{v\sigma^4}(y\_i-\mu)^2]\Big\}/\Big\{ [1+\frac{1}{v\sigma^2}(y\_i-\mu)^2]^4\Big\}.....\text{really messy!} \\ &\frac{\partial}{\partial (\sigma^2)^2}logf(y\_i)=\frac{1}{2\sigma^4}-\frac{(v+1)}{2}\frac{\frac{1}{v\sigma^6}(y\_i-\mu)^2}{[1+\frac{1}{v\sigma^2}(y\_i-\mu)^2]^2}
\end{align\*}

Finally, the **expected fisher's information matrix** is calculated as follows:
\begin{align\*}
\boldsymbol{I}= -\mathbb{E}\Big(\begin{bmatrix}
\frac{\partial^2}{\partial \mu^2}logf(y\_i) & \frac{\partial}{\partial \mu \partial \sigma^2}logf(y\_i) \\ 
\frac{\partial}{\partial \mu \partial \sigma^2}logf(y\_i) & \frac{\partial^2}{\partial (\sigma^2)^2}logf(y\_i)
\end{bmatrix}\Big)
\end{align\*}

However, I have no idea how to calculate these expectations. Is anyone aware of a resource that has done this? **Honestly, the only quantity I'm interested in is: $-\mathbb{E}\big[\frac{\partial^2}{\partial (\sigma^2)^2}logf(y\_i)\big]$, would someone at least be able to help me calculate this?**