Symbolic integration on planar differential foliations

We consider the problem of symbolic integration of $\int G(x,y(x\left)\right)dx$ where G is rational and $y\left(x\right)$ is a non algebraic solution of a differential equation $y^{\prime }\left(x\right)=F(x,y(x\left)\right)$ with F rational. Substituting in the integral $y\left(x\right)$ by $y(x,h)$, the general solution of $y^{\prime }\left(x\right)=F(x,y(x\left)\right)$, we have a parametrized integral $I(x,h)$. We prove that $I(x,h)$ is, as a two variable function in $x,h$, either differentially transcendental, or, with a good parametrization in h, there exists a linear differential operator L in h with constant coefficients, called a telescoper, such that $LI(x,h)$ is rational in $x,y$ and the h derivatives of y. This notion generalizes elementary integration. We present an algorithm to compute such telescoper given a priori bound on the order ord of L and degree N of $LI(x,h)$, with complexity $\tilde{O}(N^{\omega +1}{\text{ord}}^{\omega -1}+N{\text{ord}}^{\omega +3})$. For the specific foliation $y=\ln x$, a more complete algorithm without a priori bound is presented. Oppositely, non existence of telescoper is proven for a classical planar Hamiltonian system. As an application, we present an algorithm which always finds, if they exist, the Liouvillian solutions of a planar rational vector field, given a bound large enough for some notion of complexity height.