Integrand size = 15, antiderivative size = 70 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 52, 57, 632, 210, 31} \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log (x)}{2} \]
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Rule 31
Rule 52
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(1-x)^{2/3}}{x} \, dx,x,x^3\right ) \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right ) \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\log (x)}{2}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right ) \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right ) \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {1}{6} \left (3 \left (1-x^3\right )^{2/3}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1-x^3}\right )-\log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.93 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{3} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {1}{3},1,1;2,2;x^{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi }\) | \(66\) |
pseudoelliptic | \(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{2}-\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\arctan \left (\frac {\left (1+2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-1\right )}{3}\) | \(74\) |
trager | \(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{2}+\frac {\ln \left (\frac {-211 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3126 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}-11543 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}-19749 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+1688 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5502 \left (-x^{3}+1\right )^{\frac {1}{3}}+15935 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+21437}{x^{3}}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-1649 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+9683 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}+1266 x^{3}+19749 \left (-x^{3}+1\right )^{\frac {2}{3}}+14247 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+13192 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5502 \left (-x^{3}+1\right )^{\frac {1}{3}}-6557 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1055}{x^{3}}\right )}{3}\) | \(257\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=- \frac {x^{2} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {1}{x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 0.45 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.30 \[ \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx=\frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {{\left (1-x^3\right )}^{2/3}}{2} \]
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