Optimal. Leaf size=87 \[ -\frac {\left (x^2+1\right )^{2/3}}{2 x^2}+\frac {1}{3} \log \left (\sqrt [3]{x^2+1}-1\right )-\frac {1}{6} \log \left (\left (x^2+1\right )^{2/3}+\sqrt [3]{x^2+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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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, 47, 55, 618, 204, 31} \begin {gather*} -\frac {\left (x^2+1\right )^{2/3}}{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 47
Rule 55
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1+x^2\right )^{2/3}}{2 x^2}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^2\right )\\ &=-\frac {\left (1+x^2\right )^{2/3}}{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 {\left (1+x^2\right )^{2/3}}{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 {\left (1+x^2\right )^{2/3}}{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}{10} \left (x^2+1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 87, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^2\right )^{2/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^2}\right )-\frac {1}{6} \log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 79, normalized size = 0.91 \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 {2}{3}}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, 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 {2}{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 = 1.70, size = 76, normalized size = 0.87
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{2} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+2 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{2}}\right )}{6 \pi }\) | \(76\) |
risch | \(-\frac {\left (x^{2}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{2} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+2 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi }\) | \(76\) |
trager | \(-\frac {\left (x^{2}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{2}+1\right )^{\frac {2}{3}}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}}+36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-2 x^{2}-9 \left (x^{2}+1\right )^{\frac {1}{3}}+57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-5}{x^{2}}\right )-\frac {\ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \left (x^{2}+1\right )^{\frac {2}{3}}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}}-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+15 x^{2}+24 \left (x^{2}+1\right )^{\frac {1}{3}}+33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+20}{x^{2}}\right )}{3}-\ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \left (x^{2}+1\right )^{\frac {2}{3}}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}}-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+15 x^{2}+24 \left (x^{2}+1\right )^{\frac {1}{3}}+33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+20}{x^{2}}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(439\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, 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 {2}{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 = 0.95, size = 86, normalized size = 0.99 \begin {gather*} \frac {\ln \left ({\left (x^2+1\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (x^2+1\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (x^2+1\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {{\left (x^2+1\right )}^{2/3}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.90, size = 34, normalized size = 0.39 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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