<s>[INST] Finding $\int\frac{x^2-1}{\sqrt{x^4+x^2+1}}$ Finding
$$\int\frac{x^2-1}{\sqrt{x^4+x^2+1}}$$
Try: I have tried it to convert it into $\displaystyle \left(x+\frac{1}{x}\right)=t$ and $\displaystyle \left(1-\frac{1}{x^2}\right)$  but nothing happen.
I felt that it must be in Elliptical Integral of first kind, second kind or in third  kind.
Can someone explain me how to write in elliptical form
 [/INST] Maple expresses it as
$$ {\frac { \left( i\sqrt {3}+1 \right) \sqrt {4+2\,{x}^{2}-2\,i{x}
^{2}\sqrt {3}}\sqrt {4+2\,{x}^{2}+2\,i{x}^{2}\sqrt {3}} \left( i{\it 
EllipticF} \left(  \left( i\sqrt {3}+1 \right) x/2,(1+i\sqrt {3})/2
 \right) \sqrt {3}+3\,{\it EllipticF} \left( \left( i\sqrt {3}+1
 \right) x/2,(1+i\sqrt {3})/2 \right) -2\,{\it EllipticE} \left(
 \left( i\sqrt {3}+1 \right) x/2,(1+i\sqrt {3})/2 \right)  \right) }{16\;
\sqrt {x^4+x^2+1}}}
$$
So yes, it is expressed in terms of elliptic integrals of the first and second kinds (EllipticF is first kind, EllipticE is second kind).
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