Integrand size = 15, antiderivative size = 55 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 57, 632, 210, 31} \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log (x)}{2} \]
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Rule 31
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right ) \\ & = -\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 {\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 {\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 = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\sqrt [3]{1-x^3}\right )-\frac {1}{6} \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]
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Time = 2.00 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\arctan \left (\frac {\left (1+2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-1\right )}{3}\) | \(63\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{3} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {4}{3};2,2;x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi }\) | \(65\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-1438 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9855 \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}}+1477 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+11504 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-14247 \left (-x^{3}+1\right )^{\frac {1}{3}}+17006 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2743}{x^{3}}\right )}{3}-\frac {\ln \left (-\frac {1438 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6979 \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}}-9894 x^{3}+19749 \left (-x^{3}+1\right )^{\frac {2}{3}}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-11504 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+19749 \left (-x^{3}+1\right )^{\frac {1}{3}}-6002 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+8245}{x^{3}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\ln \left (-\frac {1438 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6979 \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}}-9894 x^{3}+19749 \left (-x^{3}+1\right )^{\frac {2}{3}}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-11504 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+19749 \left (-x^{3}+1\right )^{\frac {1}{3}}-6002 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+8245}{x^{3}}\right )}{3}\) | \(372\) |
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Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=- \frac {e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {1}{x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 0.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx=\frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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