Is my solution to this first order ODE correct? (Integrating factor method)

The question wants me to prove the following ODE via integrating factor method:
$$ frac{dx}{dt} + frac{t}{(1+t^2)}x = frac{1}{t(1+t^2)} $$

I got my integrating factor to be $ mu(t)= sqrt{1+t^2}$
and then it gets messy and my implicit solution is as follows:

$ sqrt{1+t^2}x = frac{1}{2}ln{frac{sqrt{1+t^2}-1}{sqrt{1+t^2}+1}} + C $

details of the workings are shown:

working

I’m not too sure whether I’m correct, but my peer told me that I could use some substitution $x=tan{t} $ but I’m not sure why I need to do that substitution

And also if I am actually correct, the solution to that ODE could have a totally different expression if I were to use that tangent substitution right? (and that both expressions could be correct right? How do we verify if the expressions are equivalent?)