Integrand size = 33, antiderivative size = 245 \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\frac {1}{2} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \]
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\[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{\left (-1+x^3\right )^{3/4}}+\frac {x \left (-1+2 x^3-4 x^5-x^6\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}\right ) \, dx \\ & = \int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x \left (-1+2 x^3-4 x^5-x^6\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \\ & = \frac {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\int \left (-\frac {x}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}+\frac {2 x^4}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}-\frac {4 x^6}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}-\frac {x^7}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}\right ) \, dx \\ & = \frac {x^2 \left (1-x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},x^3\right )}{2 \left (-1+x^3\right )^{3/4}}+2 \int \frac {x^4}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-4 \int \frac {x^6}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-\int \frac {x}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-\int \frac {x^7}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \\ \end{align*}
Time = 3.14 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.89 \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\frac {1}{2} \left (\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.64 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.88
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) \sqrt {x^{3}-1}\, x^{2}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} \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}^{8}+1\right )^{3}}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{9} x^{4}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \sqrt {x^{3}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (\frac {2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{2}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{3}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}\) | \(460\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (-1\right )^{\frac {1}{8}} x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {\left (-1\right )^{\frac {1}{8}} x - {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (-1\right )^{\frac {1}{8}} \log \left (\frac {i \, \left (-1\right )^{\frac {1}{8}} x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-i \, \left (-1\right )^{\frac {1}{8}} x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\int \frac {x^{6} \left (x^{3} - 4\right )}{\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{8} + x^{6} - 2 x^{3} + 1\right )}\, dx \]
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\[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{6}}{{\left (x^{8} + x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{6}}{{\left (x^{8} + x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\int \frac {x^6\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{3/4}\,\left (x^8+x^6-2\,x^3+1\right )} \,d x \]
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