Integrand size = 54, antiderivative size = 107 \[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\frac {\sqrt {1+3 x^2+x^4}}{4 \left (1-x+x^2\right )}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1-x+x^2+\sqrt {1+3 x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )}{2 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.38 (sec) , antiderivative size = 5419, normalized size of antiderivative = 50.64, number of steps used = 136, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6820, 6860, 1113, 6874, 1740, 1755, 12, 1261, 738, 210, 1730, 1203, 1149, 1228, 1470, 553, 1738, 2} \[ \text {Too large to display} \]
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Rule 2
Rule 12
Rule 210
Rule 553
Rule 738
Rule 1113
Rule 1149
Rule 1203
Rule 1228
Rule 1261
Rule 1470
Rule 1730
Rule 1738
Rule 1740
Rule 1755
Rule 6820
Rule 6860
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+x-x^5+x^6}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx \\ & = \int \left (\frac {1}{\sqrt {1+3 x^2+x^4}}+\frac {1+x}{2 \left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {-8+x}{4 \left (1-x+x^2\right ) \sqrt {1+3 x^2+x^4}}+\frac {-2-x}{4 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx \\ & = \frac {1}{4} \int \frac {-8+x}{\left (1-x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \int \frac {1+x}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx \\ & = \frac {\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 {1}{4} \int \left (\frac {1+5 i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}}+\frac {1-5 i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx+\frac {1}{4} \int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx+\frac {1}{2} \int \left (\frac {1}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {x}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}}\right ) \, dx \\ & = \frac {\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 {1}{2} \int \frac {1}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \int \frac {x}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \left (1-5 i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \left (1+5 i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx \\ & = \frac {\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 {1}{2} \int \left (-\frac {2 \left (1+i \sqrt {3}\right )}{3 \left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {2 i}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt {1+3 x^2+x^4}}-\frac {2 \left (1-i \sqrt {3}\right )}{3 \left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {2 i}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx+\frac {1}{2} \int \left (-\frac {4}{3 \left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt {1+3 x^2+x^4}}-\frac {4}{3 \left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {x}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \left (1-i \sqrt {3}\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}{2} \left (1+i \sqrt {3}\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}{2} \left (1+i \sqrt {3}\right ) \int \frac {x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \left (7-3 i \sqrt {3}\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}{2} \left (7+3 i \sqrt {3}\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}{2} \left (-1-5 i \sqrt {3}\right ) \int \frac {x}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \left (-1+5 i \sqrt {3}\right ) \int \frac {x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx \\ & = \frac {\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 {2}{3} \int \frac {1}{\left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+3 x^2+x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\frac {i \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \sqrt {3}}+\frac {(2 i) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \sqrt {3}}+\frac {(2 i) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \sqrt {3}}+\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\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 (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (-1-5 i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (-1+5 i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )+\frac {\left (7+3 i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{4 \left (2-i \sqrt {3}-\sqrt {5}\right )}+\frac {\left (7+3 i \sqrt {3}\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}{2 \left (2-i \sqrt {3}-\sqrt {5}\right )}+\frac {\left (7-3 i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{4 \left (2+i \sqrt {3}-\sqrt {5}\right )}+\frac {\left (7-3 i \sqrt {3}\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}{2 \left (2+i \sqrt {3}-\sqrt {5}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{4 \left (2 i-\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (i+\sqrt {3}\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}{2 \left (2 i-\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{4 \left (2 i+\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (i-\sqrt {3}\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}{2 \left (2 i+\sqrt {3}-i \sqrt {5}\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 1.63 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\frac {\sqrt {1+3 x^2+x^4}}{4 \left (1-x+x^2\right )}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1-x+x^2+\sqrt {1+3 x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )}{2 \sqrt {2}} \]
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Time = 4.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {\left (\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )\right ) \sqrt {2}}{8}\) | \(84\) |
default | \(\frac {\left (x^{2}-x +1\right ) \left (\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )\right ) \sqrt {2}+2 \sqrt {x^{4}+3 x^{2}+1}}{8 x^{2}-8 x +8}\) | \(94\) |
pseudoelliptic | \(\frac {\left (x^{2}-x +1\right ) \left (\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )\right ) \sqrt {2}+2 \sqrt {x^{4}+3 x^{2}+1}}{8 x^{2}-8 x +8}\) | \(94\) |
trager | \(\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{10}-29 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{9}+10 \sqrt {x^{4}+3 x^{2}+1}\, x^{8}-89 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{8}+40 \sqrt {x^{4}+3 x^{2}+1}\, x^{7}-190 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{7}+116 \sqrt {x^{4}+3 x^{2}+1}\, x^{6}-283 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{6}+192 \sqrt {x^{4}+3 x^{2}+1}\, x^{5}-363 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{5}+270 x^{4} \sqrt {x^{4}+3 x^{2}+1}-283 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}+192 \sqrt {x^{4}+3 x^{2}+1}\, x^{3}-190 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+116 \sqrt {x^{4}+3 x^{2}+1}\, x^{2}-89 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+40 \sqrt {x^{4}+3 x^{2}+1}\, x -29 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +10 \sqrt {x^{4}+3 x^{2}+1}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )^{4}}\right )}{8}\) | \(320\) |
elliptic | \(-\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 )+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 )\right )}{12 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (1+\frac {x^{2}-1}{-x^{2}-1}\right )^{2}}}\, \left (1+\frac {x^{2}-1}{-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 )}{24 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (1+\frac {x^{2}-1}{-x^{2}-1}\right )^{2}}}\, \left (1+\frac {x^{2}-1}{-x^{2}-1}\right ) \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+3\right )}+\frac {\left (\frac {1}{8+\frac {4 \sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}}-\frac {5 \ln \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}+\frac {1}{-8+\frac {4 \sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}}+\frac {5 \ln \left (-1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}\right ) \sqrt {2}}{2}\) | \(661\) |
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.72 \[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\frac {\sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (\frac {3 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} - x + 1\right )} + 9 \, x^{2} - 2 \, x + 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + 4 \, \sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (\frac {3 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} + 9 \, x^{2} + 2 \, x + 3}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + 4 \, \sqrt {x^{4} + 3 \, x^{2} + 1}}{16 \, {\left (x^{2} - x + 1\right )}} \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{\left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right ) \sqrt {x^{4} + 3 x^{2} + 1}}\, dx \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\int { \frac {{\left (x^{4} - x^{3} + x^{2} - x + 1\right )} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\int { \frac {{\left (x^{4} - x^{3} + x^{2} - x + 1\right )} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx=\int \frac {\left (x^2-1\right )\,\left (x^4-x^3+x^2-x+1\right )}{{\left (x^2-x+1\right )}^2\,\sqrt {x^4+3\,x^2+1}\,\left (x^2+x+1\right )} \,d x \]
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