Consider the standard Heston model\begin{align\*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align\*}
Computing $\mathbb{E}\int\_0^t v\_sds$ is simple but does anyone have a reference for
\begin{align}
Var\left(\int\_0^t v\_sds\right)
\end{align}
or is there a simple trick to solve this integral and compute its second moment?

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Using Ito's Lemma ($d(tv)=vdt+tdv$) and using the SDE for $dv$, I only got
\begin{align}
\int\_0^t v\_udu=tv\_t-\frac{1}{2}\kappa\theta t^2+\kappa\int\_0^t u v\_udu-\xi\int\_0^t u\sqrt{v\_u}dW\_u,
\end{align}
which doesn't look too helpful.