I believe if $k=0$ the following integral is reducible to an elliptic integral. If $k > 0$ is it possible to reduce it to an elliptic integral or some other special function?
$$\int_\rho^x \sqrt{1 + (\alpha \cos(t) - k)^2}\, dt$$
2026-03-25 14:27:22.1774448842
Is this integral reducible to an elliptic integral?
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According to Maple, it can be done... $\int \sqrt{1+(a \cos t - k)^2}\,dt$ is reported as