Integrand size = 13, antiderivative size = 92 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=-\frac {\left (-1+x^3\right )^{2/3}}{3 x^3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \log \left (1+\sqrt [3]{-1+x^3}\right )+\frac {1}{9} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 43, 58, 632, 210, 31} \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=-\frac {2 \arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (x^3-1\right )^{2/3}}{3 x^3}-\frac {1}{3} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {\log (x)}{3} \]
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Rule 31
Rule 43
Rule 58
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^2} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{3 x^3}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{3 x^3}+\frac {\log (x)}{3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{3 x^3}+\frac {\log (x)}{3}-\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right ) \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{3 x^3}-\frac {2 \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\log (x)}{3}-\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {1}{9} \left (-\frac {3 \left (-1+x^3\right )^{2/3}}{x^3}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^3}\right )+\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.49 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{3 x^{3}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(96\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{9 \pi {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}\) | \(97\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{3}-2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) x^{3}+\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) x^{3}-3 \left (x^{3}-1\right )^{\frac {2}{3}}}{9 \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}\) | \(107\) |
trager | \(-\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{3 x^{3}}-\frac {2 \ln \left (\frac {-13504 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{3}-21632 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{3}+44016 \left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-8628 x^{3}-19749 \left (x^{3}-1\right )^{\frac {2}{3}}+157992 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+108032 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+100472 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+7190}{x^{3}}\right )}{9}+\frac {2 \ln \left (-\frac {-105536 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{3}+103848 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{3}+44016 \left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-10066 x^{3}+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-113976 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+844288 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+19749 \left (x^{3}-1\right )^{\frac {1}{3}}-263528 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+18694}{x^{3}}\right )}{9}-\frac {16 \ln \left (-\frac {-105536 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{3}+103848 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{3}+44016 \left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-10066 x^{3}+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-113976 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+844288 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+19749 \left (x^{3}-1\right )^{\frac {1}{3}}-263528 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+18694}{x^{3}}\right ) \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{9}\) | \(429\) |
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Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{3} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{3} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{9 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.35 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} + \frac {1}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} + \frac {1}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 6.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=-\frac {2\,\ln \left (\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}+\frac {4}{9}\right )}{9}-\ln \left (9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )-\frac {{\left (x^3-1\right )}^{2/3}}{3\,x^3} \]
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