Integrand size = 36, antiderivative size = 98 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {\left (1+2 x+x^2\right ) \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {1+3 x^2+x^4}}{x^2}+\frac {2 \sqrt {1+3 x^2+x^4}}{x}+\frac {(-1-2 x) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2}-\frac {2 x \sqrt {1+3 x^2+x^4}}{1+x+x^2}\right ) \, dx \\ & = 2 \int \frac {\sqrt {1+3 x^2+x^4}}{x} \, dx-2 \int \frac {x \sqrt {1+3 x^2+x^4}}{1+x+x^2} \, dx-\int \frac {\sqrt {1+3 x^2+x^4}}{x^2} \, dx+\int \frac {(-1-2 x) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2} \, dx \\ & = \frac {\sqrt {1+3 x^2+x^4}}{x}-2 \int \left (\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {1+3 x^2+x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {1+3 x^2+x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx-\int \frac {3+2 x^2}{\sqrt {1+3 x^2+x^4}} \, dx+\int \left (-\frac {\sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2}-\frac {2 x \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2}\right ) \, dx+\text {Subst}\left (\int \frac {\sqrt {1+3 x+x^2}}{x} \, dx,x,x^2\right ) \\ & = \sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}-\frac {1}{2} \text {Subst}\left (\int \frac {-2-3 x}{x \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )-2 \int \frac {x^2}{\sqrt {1+3 x^2+x^4}} \, dx-2 \int \frac {x \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2} \, dx-3 \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{3} \left (2 \left (3-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{1+i \sqrt {3}+2 x} \, dx-\frac {1}{3} \left (2 \left (3+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{1-i \sqrt {3}+2 x} \, dx-\int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2} \, dx \\ & = -\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {1+3 x^2+x^4}}-\frac {3 \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1+3 x^2+x^4}}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+3 x+x^2}} \, dx,x,x^2\right )-2 \int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}-\frac {2 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}-\frac {2 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx-\frac {1}{3} \left (4 \left (3-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{3} \left (4 \left (3-i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{3} \left (4 \left (3+i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+3 x^2+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{3} \left (4 \left (3+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx-\int \left (-\frac {4 \sqrt {1+3 x^2+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \sqrt {1+3 x^2+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x \sqrt {1+3 x+x^2}} \, dx,x,x^2\right ) \\ & = -\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {1+3 x^2+x^4}}-\frac {3 \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1+3 x^2+x^4}}+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+3 x^2}{\sqrt {1+3 x^2+x^4}}\right )+3 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {3+2 x^2}{\sqrt {1+3 x^2+x^4}}\right )-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{3} \left (2 \left (3-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1+3 x+x^2}}{\left (1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{3} \left (8 \left (3-i \sqrt {3}\right )\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{12} \left (-3+i \sqrt {3}\right ) \int \frac {12+\left (1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {12+\left (1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{3} \left (2 \left (3+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1+3 x+x^2}}{\left (1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{3} \left (8 \left (3+i \sqrt {3}\right )\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx \\ & = -\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\sqrt {1+3 x^2+x^4}-\frac {1}{6} \left (3-i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}-\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}+\frac {3}{2} \text {arctanh}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\text {arctanh}\left (\frac {2+3 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {1+3 x^2+x^4}}-\frac {3 \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1+3 x^2+x^4}}+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{12} \left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {2 \left (1+3 i \sqrt {3}\right )+4 \left (2+i \sqrt {3}\right ) x}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (-3+i \sqrt {3}\right ) \int \frac {x^2}{\sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {2 \left (1-3 i \sqrt {3}\right )+4 \left (2-i \sqrt {3}\right ) x}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (3+i \sqrt {3}\right ) \int \frac {x^2}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{3} \left (2 \left (3-2 i \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{3} \left (2 \left (3+2 i \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {\left (4 \left (3 i+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i-\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (8 \left (3 i+\sqrt {3}\right )\right ) \int \frac {3-\sqrt {5}+2 x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i-\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (4 \left (3 i-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i+\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (8 \left (3 i-\sqrt {3}\right )\right ) \int \frac {3-\sqrt {5}+2 x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i+\sqrt {3}-i \sqrt {5}\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.85 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {(1+x)^2 \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \]
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Time = 6.48 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-2 \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right )+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}\) | \(77\) |
default | \(\frac {3 x \sqrt {2}\, \left (x^{2}+x +1\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+2 \left (1+x \right )^{2} \sqrt {x^{4}+3 x^{2}+1}-4 x \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right ) \left (x^{2}+x +1\right )}{2 x \left (x^{2}+x +1\right )}\) | \(91\) |
pseudoelliptic | \(\frac {3 x \sqrt {2}\, \left (x^{2}+x +1\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+2 \left (1+x \right )^{2} \sqrt {x^{4}+3 x^{2}+1}-4 x \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right ) \left (x^{2}+x +1\right )}{2 x \left (x^{2}+x +1\right )}\) | \(91\) |
trager | \(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{4}+3 x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}+x +1}\right )}{2}-2 \ln \left (\frac {1+x^{2}+\sqrt {x^{4}+3 x^{2}+1}}{x}\right )\) | \(118\) |
elliptic | \(-\ln \left (x^{2}+\frac {3}{2}+\sqrt {x^{4}+3 x^{2}+1}\right )-\operatorname {arctanh}\left (\frac {3 x^{2}+2}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\frac {2 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+6 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )+12 \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\right )}{6 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}\, \left (\frac {x^{2}-1}{-x^{2}-1}+1\right ) \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+3\right )}-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )\right )}{3 \left (\frac {x^{2}-1}{-x^{2}-1}+1\right ) \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}}+\frac {\left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}-\frac {1}{2 \left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}-1\right )}+\frac {3 \ln \left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}-1\right )}{2}-\frac {1}{2 \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}-\frac {3 \ln \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(724\) |
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Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.53 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {3 \, \sqrt {2} {\left (x^{3} + x^{2} + x\right )} \log \left (\frac {3 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} - x + 1\right )} + 9 \, x^{2} - 2 \, x + 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + 8 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-\frac {x^{2} - \sqrt {x^{4} + 3 \, x^{2} + 1} + 1}{x}\right ) + 4 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 2 \, x + 1\right )}}{4 \, {\left (x^{3} + x^{2} + x\right )}} \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 3 x^{2} + 1}}{x^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 1\right )} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 1\right )} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (x^2-1\right )\,\left (x^2+1\right )\,\sqrt {x^4+3\,x^2+1}}{x^2\,{\left (x^2+x+1\right )}^2} \,d x \]
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