Integrand size = 15, antiderivative size = 58 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 58, 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} \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log (x)}{2} \]
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Rule 31
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right ) \\ & = -\frac {\log (x)}{2}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right ) \\ & = -\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right ) \\ & = \frac {1}{2} \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {1}{4} \left (2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1-x^2}\right )-\log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \]
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Time = 1.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+\left (-x^{2}+1\right )^{\frac {1}{3}}+1\right )}{4}+\frac {\arctan \left (\frac {\left (1+2 \left (-x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2}+\frac {\ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-1\right )}{2}\) | \(63\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{2} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {4}{3};2,2;x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{4 \pi }\) | \(65\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+24 \left (-x^{2}+1\right )^{\frac {2}{3}}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-12 x^{2}+24 \left (-x^{2}+1\right )^{\frac {1}{3}}+11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+20}{x^{2}}\right )}{2}-\frac {\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9 \left (-x^{2}+1\right )^{\frac {2}{3}}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+10 x^{2}-9 \left (-x^{2}+1\right )^{\frac {1}{3}}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-5}{x^{2}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{2}-\frac {\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9 \left (-x^{2}+1\right )^{\frac {2}{3}}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+10 x^{2}-9 \left (-x^{2}+1\right )^{\frac {1}{3}}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-5}{x^{2}}\right )}{2}\) | \(370\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=- \frac {e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {1}{x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.60 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx=\frac {\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {9}{4}\right )}{2}+\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-9\,{\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-9\,{\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right ) \]
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