  Let $k$ be a differential field of characteristic zero and the field of
constants $C$ of $k$ be an algebraically closed field. Let $E$ be a
differential field extension of $k$ having $C$ as its field of constants and
that $E=E_m\supseteq E_{m-1}\supseteq\cdots\supseteq E_1\supseteq E_0=k,$ 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 special
linear group $\mathrm{SL}_2(C)$ or the infinite dihedral subgroup
$\mathrm{D}_\infty$ of $\mathrm{SL}_2(C).$ In this article, we prove that
Liouville'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 a
positive 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
$$\eta'=\sum^n_{i=1}c_i\frac{u'_i}{u_i}+v'.$$
