Integrand size = 36, antiderivative size = 254 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\frac {\sqrt {1-x-x^2+x^3+x^4}}{x}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )+\text {RootSum}\left [9+12 \text {$\#$1}+62 \text {$\#$1}^2-4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {9 \log (x)-9 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right )-10 \log (x) \text {$\#$1}+10 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right ) \text {$\#$1}^2}{3+31 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2}+\frac {2 \sqrt {1-x-x^2+x^3+x^4}}{1+x^2}\right ) \, dx \\ & = 2 \int \frac {\sqrt {1-x-x^2+x^3+x^4}}{1+x^2} \, dx-\int \frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2} \, dx \\ & = 2 \int \left (\frac {i \sqrt {1-x-x^2+x^3+x^4}}{2 (i-x)}+\frac {i \sqrt {1-x-x^2+x^3+x^4}}{2 (i+x)}\right ) \, dx-\int \frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2} \, dx \\ & = i \int \frac {\sqrt {1-x-x^2+x^3+x^4}}{i-x} \, dx+i \int \frac {\sqrt {1-x-x^2+x^3+x^4}}{i+x} \, dx-\int \frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2} \, dx \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\frac {1}{2} \left (\frac {2 \sqrt {1-x-x^2+x^3+x^4}}{x}+\log (x)-\log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )-\text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log (x)+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}-8 \log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ]\right ) \]
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Time = 8.59 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.49
method | result | size |
risch | \(\frac {\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{x}+\frac {-4 \left (\left (x^{2}-3 x -1\right ) \sqrt {13}+13 x \right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\frac {\sqrt {-2+\sqrt {13}}\, \arctan \left (\frac {35 \sqrt {-2+\sqrt {13}}\, \sqrt {13}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (-\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \left (\left (x^{2}+2 x -1\right ) \sqrt {13}+5 x^{2}-2 x -5\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{3}+\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (2 x^{2}+x -2\right )}{3 x}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, x \right )}{2 \sqrt {78+26 \sqrt {13}}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, x}\) | \(378\) |
default | \(\frac {104 \left (\left (x^{2}-\frac {13}{2} x -1\right ) \sqrt {13}-\frac {7 x^{2}}{2}+\frac {47 x}{2}+\frac {7}{2}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\sqrt {-2+\sqrt {13}}\, \arctan \left (\frac {35 \sqrt {-2+\sqrt {13}}\, \sqrt {13}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (-\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \left (\left (3 x^{2}-22 x -3\right ) \sqrt {13}-13 x^{2}+78 x +13\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}+7 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (2 x^{2}+x -2\right )}{3 x}\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\right ) \left (\sqrt {13}-\frac {26}{7}\right )\right )}{2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \sqrt {78+26 \sqrt {13}}\, x \left (7 \sqrt {13}-26\right )}\) | \(399\) |
pseudoelliptic | \(\frac {104 \left (\left (x^{2}-\frac {13}{2} x -1\right ) \sqrt {13}-\frac {7 x^{2}}{2}+\frac {47 x}{2}+\frac {7}{2}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\sqrt {-2+\sqrt {13}}\, \arctan \left (\frac {35 \sqrt {-2+\sqrt {13}}\, \sqrt {13}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (-\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \left (\left (3 x^{2}-22 x -3\right ) \sqrt {13}-13 x^{2}+78 x +13\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}+7 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (2 x^{2}+x -2\right )}{3 x}\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\right ) \left (\sqrt {13}-\frac {26}{7}\right )\right )}{2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \sqrt {78+26 \sqrt {13}}\, x \left (7 \sqrt {13}-26\right )}\) | \(399\) |
trager | \(\frac {\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) \ln \left (-\frac {-51 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{5} x^{2}+51 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{5}-362 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}-136 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}+448 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+362 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}-247 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) x^{2}-104 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) x +1872 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+247 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )}{{\left (x \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+5 x -1\right )}^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \ln \left (\frac {51 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{4} x^{2}-51 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{4} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )+250 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} x^{2}-136 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} x +448 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-250 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )-89 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) x^{2}-712 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) x +816 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+89 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )}{{\left (x \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+x -5\right )}^{2}}\right )}{2}+\frac {\ln \left (-\frac {2 x^{2}+2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+x -2}{x}\right )}{2}\) | \(648\) |
elliptic | \(\text {Expression too large to display}\) | \(286046\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.42 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {\sqrt {8 i - 12} x \log \left (\frac {\sqrt {8 i - 12} {\left (\left (73 i + 14\right ) \, x^{4} + \left (66 i - 112\right ) \, x^{3} - \left (78 i + 104\right ) \, x^{2} - \left (66 i - 112\right ) \, x + 73 i + 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (33 i - 56\right ) \, x^{2} + \left (112 i + 66\right ) \, x + 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) + \sqrt {-8 i - 12} x \log \left (\frac {\sqrt {-8 i - 12} {\left (-\left (73 i - 14\right ) \, x^{4} - \left (66 i + 112\right ) \, x^{3} + \left (78 i - 104\right ) \, x^{2} + \left (66 i + 112\right ) \, x - 73 i + 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (33 i + 56\right ) \, x^{2} - \left (112 i - 66\right ) \, x - 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \sqrt {-8 i - 12} x \log \left (\frac {\sqrt {-8 i - 12} {\left (\left (73 i - 14\right ) \, x^{4} + \left (66 i + 112\right ) \, x^{3} - \left (78 i - 104\right ) \, x^{2} - \left (66 i + 112\right ) \, x + 73 i - 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (33 i + 56\right ) \, x^{2} - \left (112 i - 66\right ) \, x - 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \sqrt {8 i - 12} x \log \left (\frac {\sqrt {8 i - 12} {\left (-\left (73 i + 14\right ) \, x^{4} - \left (66 i - 112\right ) \, x^{3} + \left (78 i + 104\right ) \, x^{2} + \left (66 i - 112\right ) \, x - 73 i - 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (33 i - 56\right ) \, x^{2} + \left (112 i + 66\right ) \, x + 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - 4 \, x \log \left (-\frac {2 \, x^{2} + x + 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} - 2}{x}\right ) - 8 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}{8 \, x} \]
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Not integrable
Time = 2.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}{x^{2} \left (x^{2} + 1\right )}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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Not integrable
Time = 6.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x^4+x^3-x^2-x+1}}{x^2\,\left (x^2+1\right )} \,d x \]
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