Weierstrass elliptic function, as $\wp^{-1}\left(z\right)$ is a multivalued function[^5]. The values of $\wp^{-1}$ in eq. have then to be chosen appropriately in order to yield the correct result [as explained, e.g., in @hoggatt_inverse_1955]. As an alternative, it is possible to compute directly the integral in eq. in terms of Legendre elliptic integrals using known formulæ [e.g., @gradshtein_table_2007 §3.131 and §3.138].

The solution for the third coordinate $\phi$ can now be computed directly by integrating eq. with respect to $\tau$: $$\int_{\phi_0}^\phi du = p_\phi\left[\int_0^\tau\frac{du}{\xi^2\left( u \right)} + \int_0^\tau\frac{du}{\eta^2\left( u \right)}\right].
\label{eq:phi_inteq}$$ It is easier to tackle the calculation via the simplified formulæ and . The integrals on the right-hand side of eq. are then in a form which can be solved through a formula involving $\wp^\prime$, $\wp^{-1}$ and the Weierstrass $\sigma$ and $\zeta$ functions (see @jules_tannery_elements_1893, chapter CXII, and @gradshtein_table_2007, §5.141.5): $$\int \frac{\wp\left( u \right) + \beta}{\gamma\wp\left( u \right) + \delta}du =
\frac{u}{\gamma}+\frac{\delta-\beta\gamma}{\gamma^2\wp^\prime\left(v\right)}\left[\ln\frac{\sigma
\left( u+v \right)}{\sigma\left( u-v \right)} - 2u\zeta\left( v \right)\right],
\label{eq:phi_integral}$$ where $v=\wp^{-1}\left(-\delta / \gamma\right)$. In this case, the multivalued character of $\wp^{-1}$ does not matter: it can be verified via the reduction formulæ of the Weierstrassian functions [@abramowitz_handbook_1964 §18.2] that *any* value of $v$ such that $\wp\left( v 