We give a self-contained, modern exposition of Édouard Goursat's 1887 theorem on pseudo-elliptic integrals -- those integrals of the form $\int F(t)\,\d t/\sqrt{R(t)}$ with $R$ a cubic or quartic polynomial that, despite living on a genus-$1$ algebraic curve, admit elementary antiderivatives. After reviewing integration in finite terms and Liouville's theorem, we present Goursat's two main theorems with proofs phrased in the language of Möbius automorphisms of the underlying hyperelliptic curve. We then develop a cube-root analog: for integrals of the form $\int F(t)\,\d t/\sqrt[3]{R(t)}$ with $R$ cubic, an order-$3$ Möbius substitution cyclically permuting the roots of $R$ induces an eigendecomposition into three pieces. Two of the three eigenpieces (eigenvalues $1$ and $ω^2$, where $ω= e^{2πi/3}$) descend through a chain of substitutions to genus-$0$ curves and yield elementary antiderivatives; the middle eigenpiece (eigenvalue $ω$) descends only to the genus-$1$ curve $y^3 = x(x-K)$ and is generically transcendental.

翻译：我们给出埃杜瓦尔·古尔萨（Édouard Goursat）1887年关于伪椭圆积分定理的自洽现代阐述。这类积分形如$\int F(t)\,\d t/\sqrt{R(t)}$，其中$R$为三次或四次多项式，尽管定义在亏格为1的代数曲线上，却允许初等反导数。在回顾有限项积分与刘维尔定理后，我们用底层超椭圆曲线上的莫比乌斯自同构语言，呈现古尔萨的两个主要定理及其证明。随后，我们发展立方根类比：对于形如$\int F(t)\,\d t/\sqrt[3]{R(t)}$（其中$R$为三次多项式）的积分，循环置换$R$根的三阶莫比乌斯替换可诱导出三分量的特征分解。其中两个特征分量（特征值为$1$和$\omega^2$，$\omega= e^{2\pi i/3}$）通过代换链降至亏格为0的曲线，并产生初等反导数；中间特征分量（特征值为$\omega$）仅降至亏格1曲线$y^3 = x(x-K)$，且一般情况下为超越函数。