integration - Integral of complete elliptic Integral of second kind

I am currently wondering wether there is an indefinite integral for

$$\int dx \;\sqrt{1-x^2} E\left (1+\frac{1}{x^2-1} \right )$$
WolframAlpha states, that there is no result in terms of analytical functions. I am not a Mathematician and don't know the current literature in the field. I wonder whether there maybe is literature more recent, which has a solution for this.

Also if there is no solution, I wonder whether there is a way to prove that there is no solution.

Looking forward to helpfull answers.

Cheers!