An integral that even wolfram integrator cannot find

Here is the ``discussion" from uwants (click me)
Another discussion from mathlinks (click me)
But I find it in a way totally different from them, that's a challenge to find the formula.

Problem. Find the explicit formula of the integral $\displaystyle \int \frac{x^2-1}{(x^2+1)\sqrt{1+x^4}}\,dx$.

My way is as follows

Have fun!