Integrand size = 13, antiderivative size = 96 \[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\frac {1}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{x^2+x^6}}\right )-\frac {1}{4} \log \left (-x^2+\sqrt [3]{x^2+x^6}\right )+\frac {1}{8} \log \left (x^4+x^2 \sqrt [3]{x^2+x^6}+\left (x^2+x^6\right )^{2/3}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2038, 2036, 335, 281, 245} \[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\frac {\sqrt {3} \sqrt [3]{x^2} \sqrt [3]{x^4+1} \arctan \left (\frac {\frac {2 \left (x^2\right )^{2/3}}{\sqrt [3]{x^4+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{x^6+x^2}}-\frac {3 \sqrt [3]{x^2} \sqrt [3]{x^4+1} \log \left (\left (x^2\right )^{2/3}-\sqrt [3]{x^4+1}\right )}{8 \sqrt [3]{x^6+x^2}} \]
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Rule 245
Rule 281
Rule 335
Rule 2036
Rule 2038
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \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*}
Time = 5.58 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x^4} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{4/3}}{x^{4/3}+2 \sqrt [3]{1+x^4}}\right )-2 \log \left (-x^{4/3}+\sqrt [3]{1+x^4}\right )+\log \left (x^{8/3}+x^{4/3} \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )\right )}{8 \sqrt [3]{x^2+x^6}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 7.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.18
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{4}\right )}{4}\) | \(17\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {-x^{2}+\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}}{x^{2}}\right )}{4}+\frac {\ln \left (\frac {x^{4}+\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}} x^{2}+\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {2}{3}}}{x^{4}}\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{2}+2 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x^{2}}\right )}{4}\) | \(94\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (149 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}-734 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-585 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x^{2}-55 x^{4}-585 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-204 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-149 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-204 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-309 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-22\right )}{4}-\frac {\ln \left (149 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+436 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+585 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x^{2}-640 x^{4}+585 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-789 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-149 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-789 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+607 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-480\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{4}+\frac {\ln \left (149 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+436 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+585 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x^{2}-640 x^{4}+585 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-789 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-149 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-789 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+607 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-480\right )}{4}\) | \(416\) |
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Time = 0.45 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\frac {1}{4} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x^{2} - \sqrt {3} {\left (539 \, x^{4} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{2205 \, x^{4} + 2197}\right ) - \frac {1}{8} \, \log \left (3 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}} + 1\right ) \]
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\[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\int \frac {x}{\sqrt [3]{x^{2} \left (x^{4} + 1\right )}}\, dx \]
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\[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\int { \frac {x}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.57 \[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{8} \, \log \left ({\left (\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{4} \, \log \left ({\left | {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 6.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {x}{\sqrt [3]{x^2+x^6}} \, dx=\frac {3\,x^2\,{\left (x^4+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -x^4\right )}{4\,{\left (x^6+x^2\right )}^{1/3}} \]
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