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Is there a way to integrate the following expression

$$ \int \frac{dt}{\cal{P}(t;g_2,g_3)-c} $$

where $\cal P$ is the Weierstrass elliptic functions and $g_2$, $g_3$, and $c$ are some (real) constants?

Thank you very much in advance.

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  • $\begingroup$ What kind of a result would you like to have? A general closed form integral function is probably too much to ask for. But if some kind of estimates are needed for the (definite or indefinite) integral, an answer might exist, if you give more specific details. $\endgroup$ Sep 8, 2014 at 20:22
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    $\begingroup$ This can always be written as a combination of ordinary integrals and elliptic integrals of the first, second and third kind, using the substitution $$dt = \frac{dp}{\sqrt{4p^3-g_2p-g_3}}.$$ $\endgroup$ Sep 8, 2014 at 20:25
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    $\begingroup$ Addendum: In fact, it turns out that the integral $$\int\frac{dt}{\mathcal{P}(t;g_2,g_3)-c}= \int\frac{dp}{(p-c)\sqrt{4p^3-g_2p-g_3}}$$ can be evaluated explicitly in terms of the incomplete elliptic integral of the third kind easily in terms of $c$ and the roots $e_1$, $e_2$, $e_3$, where $$4p^3-g_2p-g_3 = 4(p-e_1)(p-e_2)(p-e_3).$$ Check any good book on elliptic functions (or just use Maple). $\endgroup$ Sep 8, 2014 at 21:40

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