Optimal. Leaf size=96 \[ -\frac {1}{4} \log \left (\sqrt [3]{x^6+x^2}-x^2\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{x^6+x^2}}\right )+\frac {1}{8} \log \left (x^4+\sqrt [3]{x^6+x^2} x^2+\left (x^6+x^2\right )^{2/3}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2013, 2011, 329, 275, 239} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{x^2} \sqrt [3]{x^4+1} \tan ^{-1}\left (\frac {\frac {2 \left (x^2\right )^{2/3}}{\sqrt [3]{x^4+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{x^6+x^2}}-\frac {3 \sqrt [3]{x^2} \sqrt [3]{x^4+1} \log \left (\left (x^2\right )^{2/3}-\sqrt [3]{x^4+1}\right )}{8 \sqrt [3]{x^6+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 275
Rule 329
Rule 2011
Rule 2013
Rubi steps
\begin {align*} \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x+x^3}} \, dx,x,x^2\right )\\ &=\frac {\left (\sqrt [3]{x^2} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \, dx,x,x^2\right )}{2 \sqrt [3]{x^2+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x^2} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x^2}\right )}{2 \sqrt [3]{x^2+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x^2} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,\left (x^2\right )^{2/3}\right )}{4 \sqrt [3]{x^2+x^6}}\\ &=\frac {\sqrt {3} \sqrt [3]{x^2} \sqrt [3]{1+x^4} \tan ^{-1}\left (\frac {1+\frac {2 \left (x^2\right )^{2/3}}{\sqrt [3]{1+x^4}}}{\sqrt {3}}\right )}{4 \sqrt [3]{x^2+x^6}}-\frac {3 \sqrt [3]{x^2} \sqrt [3]{1+x^4} \log \left (\left (x^2\right )^{2/3}-\sqrt [3]{1+x^4}\right )}{8 \sqrt [3]{x^2+x^6}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.41 \begin {gather*} \frac {3 \left (x^6+x^2\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-x^4\right )}{4 \left (x^4+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 96, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{x^2+x^6}}\right )-\frac {1}{4} \log \left (-x^2+\sqrt [3]{x^2+x^6}\right )+\frac {1}{8} \log \left (x^4+x^2 \sqrt [3]{x^2+x^6}+\left (x^2+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 92, normalized size = 0.96 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x^{2} - \sqrt {3} {\left (539 \, x^{4} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{2205 \, x^{4} + 2197}\right ) - \frac {1}{8} \, \log \left (3 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 55, normalized size = 0.57 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{8} \, \log \left ({\left (\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{4} \, \log \left ({\left | {\left (\frac {1}{x^{4}} + 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 = 4.64, size = 17, normalized size = 0.18
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{4}\right )}{4}\) | \(17\) |
trager | \(-\frac {\ln \left (11 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}-215 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+789 \left (x^{6}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+800 x^{4}-585 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-585 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-204 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-11 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+138 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+320\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-160 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}-629 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+204 \left (x^{6}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+44 x^{4}+585 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+585 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-789 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+160 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-491 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+33\right )}{4}\) | \(275\) |
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 \begin {gather*} \int \frac {x}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 31, normalized size = 0.32 \begin {gather*} \frac {3\,x^2\,{\left (x^4+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -x^4\right )}{4\,{\left (x^6+x^2\right )}^{1/3}} \end {gather*}
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 \begin {gather*} \int \frac {x}{\sqrt [3]{x^{2} \left (x^{4} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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