Integrand size = 32, antiderivative size = 133 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3}}{4 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^6}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^6}\right )}{6\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^6}+2^{2/3} \left (1+x^6\right )^{2/3}\right )}{12\ 2^{2/3}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.56 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.81, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6860, 281, 371, 1452, 440, 476, 524} \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\frac {\left (-\sqrt {15}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{\sqrt {15}+7 i}+\frac {\left (\sqrt {15}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{-\sqrt {15}+7 i}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {8 x^6}{7-i \sqrt {15}}\right )}{7-i \sqrt {15}}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {8 x^6}{7+i \sqrt {15}}\right )}{7+i \sqrt {15}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2} \]
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Rule 281
Rule 371
Rule 440
Rule 476
Rule 524
Rule 1452
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (1+x^6\right )^{2/3}}{2 x^3}+\frac {\left (-1+4 x^3\right ) \left (1+x^6\right )^{2/3}}{2 \left (2-x^3+2 x^6\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\left (1+x^6\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+4 x^3\right ) \left (1+x^6\right )^{2/3}}{2-x^3+2 x^6} \, dx \\ & = -\left (\frac {1}{4} \text {Subst}\left (\int \frac {\left (1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\frac {1}{2} \int \left (\frac {4 \left (1+x^6\right )^{2/3}}{-1-i \sqrt {15}+4 x^3}+\frac {4 \left (1+x^6\right )^{2/3}}{-1+i \sqrt {15}+4 x^3}\right ) \, dx \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2}+2 \int \frac {\left (1+x^6\right )^{2/3}}{-1-i \sqrt {15}+4 x^3} \, dx+2 \int \frac {\left (1+x^6\right )^{2/3}}{-1+i \sqrt {15}+4 x^3} \, dx \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2}+2 \int \left (\frac {\left (i-\sqrt {15}\right ) \left (1+x^6\right )^{2/3}}{2 \left (7 i+\sqrt {15}+8 i x^6\right )}+\frac {2 x^3 \left (1+x^6\right )^{2/3}}{7-i \sqrt {15}+8 x^6}\right ) \, dx+2 \int \left (\frac {\left (-i-\sqrt {15}\right ) \left (1+x^6\right )^{2/3}}{2 \left (-7 i+\sqrt {15}-8 i x^6\right )}+\frac {2 x^3 \left (1+x^6\right )^{2/3}}{7+i \sqrt {15}+8 x^6}\right ) \, dx \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2}+4 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{7-i \sqrt {15}+8 x^6} \, dx+4 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{7+i \sqrt {15}+8 x^6} \, dx+\left (-i-\sqrt {15}\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-7 i+\sqrt {15}-8 i x^6} \, dx+\left (i-\sqrt {15}\right ) \int \frac {\left (1+x^6\right )^{2/3}}{7 i+\sqrt {15}+8 i x^6} \, dx \\ & = \frac {\left (i-\sqrt {15}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{7 i+\sqrt {15}}+\frac {\left (i+\sqrt {15}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{7 i-\sqrt {15}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2}+2 \text {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{7-i \sqrt {15}+8 x^3} \, dx,x,x^2\right )+2 \text {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{7+i \sqrt {15}+8 x^3} \, dx,x,x^2\right ) \\ & = \frac {\left (i-\sqrt {15}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{7 i+\sqrt {15}}+\frac {\left (i+\sqrt {15}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{7 i-\sqrt {15}}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {8 x^6}{7-i \sqrt {15}}\right )}{7-i \sqrt {15}}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {8 x^6}{7+i \sqrt {15}}\right )}{7+i \sqrt {15}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2} \\ \end{align*}
Time = 1.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\frac {1}{24} \left (\frac {6 \left (1+x^6\right )^{2/3}}{x^2}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^6}}\right )+2 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^6}\right )-\sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^6}+2^{2/3} \left (1+x^6\right )^{2/3}\right )\right ) \]
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Timed out.
\[\int \frac {\left (x^{6}-1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{x^{3} \left (2 x^{6}-x^{3}+2\right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (97) = 194\).
Time = 89.28 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.62 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=-\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{13} + x^{10} + 3 \, x^{7} + x^{4} + 2 \, x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (8 \, x^{18} + 60 \, x^{15} + 48 \, x^{12} + 119 \, x^{9} + 48 \, x^{6} + 60 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (4 \, x^{14} + 14 \, x^{11} + 9 \, x^{8} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{18} - 12 \, x^{15} - 24 \, x^{12} - 25 \, x^{9} - 24 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{6} - x^{3} + 2\right )} - 12 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{6} - x^{3} + 2}\right ) + 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{7} + x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{12} + 14 \, x^{9} + 9 \, x^{6} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{8} + x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{4 \, x^{12} - 4 \, x^{9} + 9 \, x^{6} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{144 \, x^{2}} \]
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Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{{\left (2 \, x^{6} - x^{3} + 2\right )} x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\int \frac {\left (x^6-1\right )\,{\left (x^6+1\right )}^{2/3}}{x^3\,\left (2\,x^6-x^3+2\right )} \,d x \]
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