Notice your four equations have two copies of

4*(a^2+a*Sqrt[a^2-1])

and the first of those is equal to a simple expression without needing to divide or square any expression. You can turn your system into three equations without square roots by substituting that simple expression into your other equation. Solve should be much faster if it does not have to deal with square roots.

If you then use Simplify on the remaining three equations you should find one equation of the form

numerator/denominator==0

and that can be replaced with

numerator==0

as long as you check after you are finished that the denominator was not equal to zero in your solutions.

That should give you three fairly simply polynomial equations and

Solve[{three resulting polynomals==0},{g1,g2,j1}]

should give you large complicated solutions in a few seconds.

It may be possible with more calculation time to Solve for all four variables, but that is less certain.

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