Optimal. Leaf size=97 \[ -\frac {1}{27} \log \left (\sqrt [3]{x^3+1}-1\right )+\frac {1}{54} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\sqrt [3]{x^3+1} \left (-x^3-3\right )}{18 x^6} \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {266, 47, 51, 57, 618, 204, 31} \begin {gather*} -\frac {\sqrt [3]{x^3+1}}{18 x^3}-\frac {1}{18} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\sqrt [3]{x^3+1}}{6 x^6}+\frac {\log (x)}{18} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 51
Rule 57
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{6 x^6}+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}-\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}+\frac {\log (x)}{18}+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}+\frac {\log (x)}{18}-\frac {1}{18} \log \left (1-\sqrt [3]{1+x^3}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\log (x)}{18}-\frac {1}{18} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.27 \begin {gather*} -\frac {1}{4} \left (x^3+1\right )^{4/3} \, _2F_1\left (\frac {4}{3},3;\frac {7}{3};x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 97, normalized size = 1.00 \begin {gather*} -\frac {1}{27} \log \left (\sqrt [3]{x^3+1}-1\right )+\frac {1}{54} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\sqrt [3]{x^3+1} \left (-x^3-3\right )}{18 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 83, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{3} + 3\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 0.79 \begin {gather*} \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} + \frac {1}{54} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 69, normalized size = 0.71 \begin {gather*} -\frac {x^{6}+4 x^{3}+3}{18 x^{6} \left (x^{3}+1\right )^{\frac {2}{3}}}-\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{27 \Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 91, normalized size = 0.94 \begin {gather*} \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{54} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 108, normalized size = 1.11 \begin {gather*} \frac {\frac {{\left (x^3+1\right )}^{1/3}}{9}+\frac {{\left (x^3+1\right )}^{4/3}}{18}}{2\,x^3-{\left (x^3+1\right )}^2+1}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{81}-\frac {1}{81}\right )}{27}-\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{3}+\frac {1}{6}-\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{3}+\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.05, size = 32, normalized size = 0.33 \begin {gather*} - \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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