trig representation

[parisse]

The best way to check an antiderivative is to run simplify(diff(anti)-f). That will remove absolute values introduced in logarithms and also eat the constant of integration.
There are several equivalent ways to write an expression, especially when inverse trig functions or ln are involved and it's not necessarily easy to move from one to another. If you know an algorithm to do so, I'd be happy to know it.
And keep in memory that the giac integration algorithm is not a table lookup in US textbooks exercices question/answers.