Wikipedia says,
There is no general closed-form solution to the three-body problem, meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations.
I tried to find the proof for it. However, while there are many results in Google saying "the three-body problem is not integrable", there is no "it has no closed-form solution" statement.
Here are some details.
According to the Picard-Vessiot theory, a field $L$ is a Liouville extension of another field $K$ if and only if, the identity component of the differential Galois group of $L/K$ is solvable. This fact is from theorem 6.1 of Teresa Crespo and Zbigniew Hajto, Introduction to Differential Galois Theory.
On the other hand, according to the Morales-Ramis theory, a Hamiltonian system is integrable if and only if, the identity component of the differential Galois group is abelian. This is theorem 1 of Morales-Ruiz, Juan J.; Ramis, Jean-Pierre; Simó, Carles, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations.
Hence, the non-existence of the closed-form solution which is algebraic over elementary formulas, would imply non-integrability (since abelianness implies solvability), but not vice versa. But as I said, there are only non-integrability results on Google...
Here are some related opinions.
There is an answer in Quora saying that the non-existence of the closed-form solution is just folklore. I'm not sure whether this is true.
In the article on Wikipedia, there is a talk, which says that the non-existence of the closed-term solution was not proved. As a result, the word "closed-form" was removed in the n-body problem subarticle, while it remained in the main article.
An article in Scholarpedia, "Three-body problem", only states about non-integrability, and does not say anything about the non-existence of the closed-form solution.
These are all I found until now. Please let me know which one is right.
