Optimal. Leaf size=144 \[ -\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {7 \left (1-x^3\right )^{2/3}}{10 x^5}+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}-\frac {4 \left (1-x^3\right )^{2/3}}{5 x^2} \]
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Rubi [C] time = 8.32, antiderivative size = 397, normalized size of antiderivative = 2.76, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {510} \[ -\frac {54 \left (x^3+1\right )^2 \left (6 x^3+1\right ) x^6 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+54 \left (x^3+1\right )^3 x^6 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+567 x^{15} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+594 x^{15} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-378 x^{12} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+972 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-819 x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+342 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+476 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-36 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+182 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-28 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-567 x^{15}+378 x^{12}+819 x^9-476 x^6-182 x^3+28}{70 x^8 \left (1-x^3\right )^{7/3}} \]
Warning: Unable to verify antiderivative.
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Rule 510
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac {28-182 x^3-476 x^6+819 x^9+378 x^{12}-567 x^{15}-28 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+182 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+476 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-819 x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-378 x^{12} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+567 x^{15} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-36 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+342 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+972 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+594 x^{15} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+54 x^6 \left (1+x^3\right )^2 \left (1+6 x^3\right ) \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+54 x^6 \left (1+x^3\right )^3 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )}{70 x^8 \left (1-x^3\right )^{7/3}}\\ \end {align*}
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Mathematica [A] time = 5.16, size = 133, normalized size = 0.92 \[ \frac {2 \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt {3}}\right )-\log \left (-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{12 \sqrt [3]{2}}-\frac {-8 x^6+x^3+2}{10 x^5 \sqrt [3]{1-x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 4.99, size = 316, normalized size = 2.19 \[ -\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{8} - x^{5}\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, \sqrt {6} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (8 \, x^{6} - x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, {\left (x^{8} - x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.02, size = 851, normalized size = 5.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^6\,{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{6} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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