Your biggest problem is you have no precision in your equations. Your equations have terms that are order of 10^0 and terms that have order 10^15. You are using machine precision numbers that only have 16 decimal places. I suggest you re-derive your equations using a higher precision (i.e. 30) and you will probably be able to integrate them more easily.

Also, in understanding the physical source of your equations, does the size of your coefficients make sense? It is unusual to have such large numbers in a dynamical equation.

Regards,

Neil

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int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]