Integrand size = 35, antiderivative size = 107 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \]
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\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-4+x^3}{2 \sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )}+\frac {-4+x^3}{2 \sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-4+x^3}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {-4+x^3}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )} \, dx \\ & = \frac {1}{2} \int \left (-\frac {4}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )}\right ) \, dx+\frac {1}{2} \int \left (-\frac {4}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {x^3}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )} \, dx+\frac {1}{2} \int \frac {x^3}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )} \, dx-2 \int \frac {1}{\sqrt [4]{-1+x^3} \left (1-x^3+x^4\right )} \, dx-2 \int \frac {1}{\sqrt [4]{-1+x^3} \left (-1+x^3+x^4\right )} \, dx \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.89 (sec) , antiderivative size = 428, normalized size of antiderivative = 4.00
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )}{2}+\frac {\ln \left (\frac {2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}-1}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}+1}{x^{4}-x^{3}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}-1}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}-1}\right )}{2}\) | \(428\) |
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Result contains complex when optimal does not.
Time = 34.74 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.56 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=-\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} - i - 1\right )}}{x^{4} + x^{3} - 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} + i + 1\right )}}{x^{4} + x^{3} - 1}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} + i - 1\right )}}{x^{4} + x^{3} - 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} - i + 1\right )}}{x^{4} + x^{3} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, {\left ({\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - 2 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - x^{3} + 1}\right ) \]
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Timed out. \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int \frac {x^4\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{1/4}\,\left (x^8-x^6+2\,x^3-1\right )} \,d x \]
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