Definite integration of Weierstrass sigma and Jacobi theta function

Question

Scrolling through QFT articles I found, not solved, the following definite integrals: $$\int_{I} d^{2}z \ln \left| \sigma(z) \right|,$$ $$\int_{I'} d^{2}z \ln \left| \theta_{1}(z) \right|,$$ $$\int_{I'} d^{2}z \ln \left| \theta_{2}(z) \right|.$$ Here $z = \alpha_{1} + i\alpha_{2}$ and $\sigma(z)$ is the Weierstrass sigma function with half-periods $\omega_{1} = R$ and $\omega_{2} = T$ and modular parameter $\tau = i\frac{T}{R}$, while $\theta_{1}(z)$ and $\theta_{2}(z)$ are the first two Jacobi theta functions with modular parameter $\tau = i\frac{T}{2R}$. The domains of integration are $I = [0,r][0,t]$ and $I' = [-\frac{r}{2}, \frac{r}{2}][-\frac{t}{2},\frac{t}{2}]$, with $t < T$ and $r < R$.
Being a simple physicist I managed to grasp their meaning in the context, but I have no clue on how to perform analytically the integration, or whether a closed form is achievable at all. Can someone help me through their computation?