Just use trig sub.
Let
Make the subs and it simplifies down to:
After you integrate cot, then make the sub to get it back in terms of x.
You're finished.
simple answer: coz you did it wrong.
split it up first: ∫√[1/x^4 + 1/x²] dx
then do the substitution, u = 1/x, dx = -1/u² du
-∫√[u^4 + u²] * 1/u² du
factor out u from inside the square root and cancel one from denominator, you get
-∫√[u²-1]/u du
you can use a trig substitution to solve this.