Optimal. Leaf size=87 \[ -\frac {\sqrt [3]{x^2+1}}{2 x^2}-\frac {1}{3} \log \left (\sqrt [3]{x^2+1}-1\right )+\frac {1}{6} \log \left (\left (x^2+1\right )^{2/3}+\sqrt [3]{x^2+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 51, 57, 618, 204, 31} \begin {gather*} -\frac {\sqrt [3]{x^2+1}}{2 x^2}-\frac {1}{2} \log \left (1-\sqrt [3]{x^2+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 57
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{1+x^2}}{2 x^2}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{1+x^2}}{2 x^2}+\frac {\log (x)}{3}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^2}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )\\ &=-\frac {\sqrt [3]{1+x^2}}{2 x^2}+\frac {\log (x)}{3}-\frac {1}{2} \log \left (1-\sqrt [3]{1+x^2}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^2}\right )\\ &=-\frac {\sqrt [3]{1+x^2}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{3}-\frac {1}{2} \log \left (1-\sqrt [3]{1+x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.30 \begin {gather*} \frac {3}{2} \sqrt [3]{x^2+1} \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 87, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [3]{x^2+1}}{2 x^2}-\frac {1}{3} \log \left (\sqrt [3]{x^2+1}-1\right )+\frac {1}{6} \log \left (\left (x^2+1\right )^{2/3}+\sqrt [3]{x^2+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 78, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 66, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 59, normalized size = 0.68 \begin {gather*} -\frac {\left (x^{2}+1\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{2}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+2 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{3 \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 = 0.57, size = 66, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} + 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 = 0.89, size = 80, normalized size = 0.92 \begin {gather*} -\frac {\ln \left ({\left (x^2+1\right )}^{1/3}-1\right )}{3}-\frac {{\left (x^2+1\right )}^{1/3}}{2\,x^2}-\ln \left (3\,{\left (x^2+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (3\,{\left (x^2+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.92, size = 32, normalized size = 0.37 \begin {gather*} - \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {10}{3}} \Gamma \left (\frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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