Integrand size = 33, antiderivative size = 120 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {1-x-x^2+x^3+x^4}}-\frac {2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx \\ & = -\left (2 \int \left (\frac {i}{2 (i-x) \sqrt {1-x-x^2+x^3+x^4}}+\frac {i}{2 (i+x) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx \\ & = -\left (i \int \frac {1}{(i-x) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )-i \int \frac {1}{(i+x) \sqrt {1-x-x^2+x^3+x^4}} \, dx+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
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Time = 4.72 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.19
method | result | size |
default | \(\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {6+2 \sqrt {13}}\, \left (\sqrt {13}-\frac {7}{2}\right ) \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\arctan \left (\frac {35 \sqrt {13}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \sqrt {-2+\sqrt {13}}\, \left (-\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \sqrt {-2+\sqrt {13}}\right )}{\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\sqrt {13}+3\right ) \left (7 \sqrt {13}-26\right ) x}\) | \(263\) |
pseudoelliptic | \(\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {6+2 \sqrt {13}}\, \left (\sqrt {13}-\frac {7}{2}\right ) \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\arctan \left (\frac {35 \sqrt {13}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \sqrt {-2+\sqrt {13}}\, \left (-\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \sqrt {-2+\sqrt {13}}\right )}{\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\sqrt {13}+3\right ) \left (7 \sqrt {13}-26\right ) x}\) | \(263\) |
trager | \(-\frac {\operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-8112 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{5} x -3536 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3} x^{2}-4147 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3} x +1820 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3536 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3}-85 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) x^{2}-95 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) x +762 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+85 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )}{{\left (13 x \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3 x -2\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \ln \left (\frac {-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{4} x +272 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} x^{2}-257 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} x +1820 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}-272 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right )+119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x +78 \sqrt {x^{4}+x^{3}-x^{2}-x +1}-119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right )}{{\left (13 x \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3 x +2\right )}^{2}}\right )}{26}\) | \(575\) |
elliptic | \(\text {Expression too large to display}\) | \(105692\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.68 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{104} \, \sqrt {13} \sqrt {-8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {-8 i - 12} {\left (\left (544 i + 731\right ) \, x^{4} + \left (1524 i - 466\right ) \, x^{3} + \left (156 i - 1586\right ) \, x^{2} - \left (1524 i - 466\right ) \, x + 544 i + 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (140 i - 171\right ) \, x^{2} + \left (342 i + 280\right ) \, x + 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{104} \, \sqrt {13} \sqrt {8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {8 i - 12} {\left (-\left (544 i - 731\right ) \, x^{4} - \left (1524 i + 466\right ) \, x^{3} - \left (156 i + 1586\right ) \, x^{2} + \left (1524 i + 466\right ) \, x - 544 i + 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (140 i + 171\right ) \, x^{2} - \left (342 i - 280\right ) \, x - 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{104} \, \sqrt {13} \sqrt {8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {8 i - 12} {\left (\left (544 i - 731\right ) \, x^{4} + \left (1524 i + 466\right ) \, x^{3} + \left (156 i + 1586\right ) \, x^{2} - \left (1524 i + 466\right ) \, x + 544 i - 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (140 i + 171\right ) \, x^{2} - \left (342 i - 280\right ) \, x - 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{104} \, \sqrt {13} \sqrt {-8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {-8 i - 12} {\left (-\left (544 i + 731\right ) \, x^{4} - \left (1524 i - 466\right ) \, x^{3} - \left (156 i - 1586\right ) \, x^{2} + \left (1524 i - 466\right ) \, x - 544 i - 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (140 i - 171\right ) \, x^{2} + \left (342 i + 280\right ) \, x + 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) \]
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Not integrable
Time = 1.84 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.24 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + 1\right )}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + 1\right )}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \]
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