Integrand size = 38, antiderivative size = 123 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \]
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\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt [4]{-1+x^2+x^4}}{x^6}+\frac {\sqrt [4]{-1+x^2+x^4}}{x^4}-\frac {4 \sqrt [4]{-1+x^2+x^4}}{x^2}+\frac {2 \sqrt [4]{-1+x^2+x^4}}{1+x^2}+\frac {4 \sqrt [4]{-1+x^2+x^4}}{-1+2 x^2}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{x^6} \, dx\right )+2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx-4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx+\int \frac {\sqrt [4]{-1+x^2+x^4}}{x^4} \, dx \\ & = 2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx+\frac {\sqrt [4]{-1+x^2+x^4} \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^4} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (2 \sqrt [4]{-1+x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^6} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (4 \sqrt [4]{-1+x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}} \\ & = \frac {4 \sqrt [4]{-1+x^2+x^4} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{4},-\frac {1}{4},\frac {1}{2},-\frac {2 x^2}{1-\sqrt {5}},-\frac {2 x^2}{1+\sqrt {5}}\right )}{x \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (1+\frac {2 x^2}{1+\sqrt {5}}\right )^{5/4} \sqrt [4]{-1+x^2+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},-\frac {1}{2},-\frac {2 \left (\frac {x^2}{1-\sqrt {5}}-\frac {x^2}{1+\sqrt {5}}\right )}{1+\frac {2 x^2}{1+\sqrt {5}}}\right )}{3 x^3 \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}}}-\frac {4 \left (1+\frac {2 x^2}{1+\sqrt {5}}\right ) \sqrt [4]{-1+x^2+x^4} \left (\left (3 \left (1+\sqrt {5}\right )-\left (13+3 \sqrt {5}\right ) x^2+2 \left (1+\sqrt {5}\right ) x^4\right ) \operatorname {Gamma}\left (-\frac {1}{4}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,-\frac {1}{2},-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )-4 x^2 \left (5+\sqrt {5}+2 \sqrt {5} x^2\right ) \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},2,\frac {1}{2},-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )\right )}{15 \left (3+\sqrt {5}\right ) x^5 \left (1-\sqrt {5}+2 x^2\right ) \operatorname {Gamma}\left (-\frac {1}{4}\right )}+2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \]
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Time = 5.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.23
method | result | size |
pseudoelliptic | \(\frac {-5 x^{5} \left (2 \arctan \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+\ln \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}+x^{2}-1}}{\sqrt {x^{4}+x^{2}-1}-\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}}\right )\right ) \sqrt {2}+4 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{10 x^{5}}\) | \(151\) |
trager | \(\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{5 x^{5}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )\) | \(239\) |
risch | \(\frac {\frac {18}{5} x^{8}+\frac {16}{5} x^{6}-\frac {18}{5} x^{4}+\frac {4}{5} x^{2}-\frac {2}{5}}{x^{5} \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{2}-1\right ) \left (x^{2}+1\right ) \left (x^{4}+x^{2}-1\right )^{2}}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{8}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{2}-1\right ) \left (x^{2}+1\right ) \left (x^{4}+x^{2}-1\right )^{2}}\right )\right ) {\left (\left (x^{4}+x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}\) | \(849\) |
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Result contains complex when optimal does not.
Time = 5.86 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.78 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {-\left (5 i + 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x - i \, x^{2} + i}{2 \, x^{4} + x^{2} - 1}\right ) + \left (5 i - 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + i \, x^{2} - i}{2 \, x^{4} + x^{2} - 1}\right ) - \left (5 i - 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + i \, x^{2} - i}{2 \, x^{4} + x^{2} - 1}\right ) + \left (5 i + 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x - i \, x^{2} + i}{2 \, x^{4} + x^{2} - 1}\right ) + 8 \, {\left (9 \, x^{4} - x^{2} + 1\right )} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}}}{20 \, x^{5}} \]
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\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - 2\right ) \sqrt [4]{x^{4} + x^{2} - 1}}{x^{6} \left (x^{2} + 1\right ) \left (2 x^{2} - 1\right )}\, dx \]
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\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )} {\left (x^{2} - 2\right )}}{{\left (2 \, x^{4} + x^{2} - 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )} {\left (x^{2} - 2\right )}}{{\left (2 \, x^{4} + x^{2} - 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int \frac {\left (x^2-1\right )\,\left (x^2-2\right )\,{\left (x^4+x^2-1\right )}^{1/4}}{x^6\,\left (2\,x^4+x^2-1\right )} \,d x \]
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