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Let $k$ be a differential field of characteristic zero and the field ofconstants $C$ of $k$ be an algebraically closed field. Let $E$ be adifferential field extension of $k$ having $C$ as its field of constants andthat $E=E_m\supseteq E_{m-1}\supseteq\cdots\supseteq E_1\supseteq E_0=k,$ where$E_i$ is either an elementary extension of $E_{i-1}$ or $E_i=E_{i-1}(t_i,t'_i)$ and $t_i$ is weierstrassian (in the sense of Kolchin ([Page 803,Kolchin1953]) over $E_{i-1}$ or $E_i$ is a Picard-Vessiot extension of$E_{i-1}$ having a differential Galois group isomorphic to either the speciallinear group $\mathrm{SL}_2(C)$ or the infinite dihedral subgroup$\mathrm{D}_\infty$ of $\mathrm{SL}_2(C).$ In this article, we prove thatLiouville's theorem on integration in finite terms ([Theorem, Rosenlicht1968])holds for $E$. That is, if $\eta\in E$ and $\eta'\in k$ then there is apositive integer $n$ and for $i=1,2,\dots,n,$ there are elements $c_i\in C,$$u_i\in k\setminus \{0\}$ and $v\in k$ such that$$\eta'=\sum^n_{i=1}c_i\frac{u'_i}{u_i}+v'.$$

Liouville's Theorem on integration in finite terms for $\mathrm D_\infty,$ $ \mathrm{SL}_2$ and Weierstrass field extensions