What did you get for your partial-fraction decomposition? Because I got
but it doesn't look like it would lead to your final step.
The procedure of integrate the above is quite standard. I don't like to use wolfram in many occasions, but this is an exception...
int 1/(X^3-1) - Wolfram|Alpha
(Press on 'show steps')
We should note that
And,
Resulting in,
Partial Fractions yields
This yields,
The second integral can be evaluated as follows,
Now let's look at
What happens when we try to make this look like the inverse tan integral?
If you expand inside the integral (to check that we have done this correctly) we will see that
It should be obvious then that,
I leave you to consider