The integral of 1/(square root of 1+x^4)?

Answer 1

That is a standard integral that has a rule, which is:

∫1/(a^2+x^2)^.5 dx = ln(x + (x^2+a^2)^.5 +c

So in your case,

∫1/(1^2+x^2^2)^.5 = ln( x^2 + (x^4+1)^.5) + c

Answer 2

Sorry. This integral is not elementary. It can be reduced to a
standard type of elliptic integral. If you
need more information about this type of
integral, run it through integrals.wolfram.com
and then click on the link about elliptic
integrals of the first kind.
In general, when you have to integrate
the square root of a cubic or quartic
polynomial, you will get an elliptic integral.