Integrand size = 15, antiderivative size = 94 \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\frac {1}{3} x \left (2+x^3\right )^{2/3}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x^3}}\right )}{3 \sqrt {3}}+\frac {5}{9} \log \left (-x+\sqrt [3]{2+x^3}\right )-\frac {5}{18} \log \left (x^2+x \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {396, 245} \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=-\frac {5 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \left (x^3+2\right )^{2/3} x+\frac {5}{6} \log \left (\sqrt [3]{x^3+2}-x\right ) \]
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Rule 245
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \left (2+x^3\right )^{2/3}-\frac {5}{3} \int \frac {1}{\sqrt [3]{2+x^3}} \, dx \\ & = \frac {1}{3} x \left (2+x^3\right )^{2/3}-\frac {5 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {5}{6} \log \left (-x+\sqrt [3]{2+x^3}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\frac {1}{18} \left (6 x \left (2+x^3\right )^{2/3}-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x^3}}\right )+10 \log \left (-x+\sqrt [3]{2+x^3}\right )-5 \log \left (x^2+x \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 4.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {x \left (x^{3}+2\right )^{\frac {2}{3}}}{3}-\frac {5 \,2^{\frac {2}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -\frac {x^{3}}{2}\right )}{6}\) | \(29\) |
meijerg | \(-\frac {2^{\frac {2}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -\frac {x^{3}}{2}\right )}{2}+\frac {2^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {x^{3}}{2}\right )}{8}\) | \(38\) |
pseudoelliptic | \(\frac {6 x \left (x^{3}+2\right )^{\frac {2}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+2\right )^{\frac {1}{3}}\right )}{3 x}\right )+10 \ln \left (\frac {-x +\left (x^{3}+2\right )^{\frac {1}{3}}}{x}\right )-5 \ln \left (\frac {x^{2}+x \left (x^{3}+2\right )^{\frac {1}{3}}+\left (x^{3}+2\right )^{\frac {2}{3}}}{x^{2}}\right )}{9 \left (-x +\left (x^{3}+2\right )^{\frac {1}{3}}\right ) \left (x^{2}+x \left (x^{3}+2\right )^{\frac {1}{3}}+\left (x^{3}+2\right )^{\frac {2}{3}}\right )}\) | \(119\) |
trager | \(\frac {x \left (x^{3}+2\right )^{\frac {2}{3}}}{3}+\frac {5 \ln \left (-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+6 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-8 x^{3} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+3 x \left (x^{3}+2\right )^{\frac {2}{3}}-4 x^{3}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-4\right )}{9}+\frac {10 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-6 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +6 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+3 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-2 x^{3}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-4\right )}{9}\) | \(226\) |
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Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\frac {1}{3} \, {\left (x^{3} + 2\right )}^{\frac {2}{3}} x + \frac {5}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {5}{9} \, \log \left (-\frac {x - {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} + 2\right )}^{\frac {1}{3}} x + {\left (x^{3} + 2\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\frac {2^{\frac {2}{3}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac {7}{3}\right )} - \frac {2^{\frac {2}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\frac {5}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {2 \, {\left (x^{3} + 2\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} + 2}{x^{3}} - 1\right )}} - \frac {5}{18} \, \log \left (\frac {{\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 2\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {5}{9} \, \log \left (\frac {{\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\int { \frac {x^{3} - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx=\int \frac {x^3-1}{{\left (x^3+2\right )}^{1/3}} \,d x \]
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