Integrand size = 18, antiderivative size = 101 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=\frac {\left (1-3 x^3\right ) \sqrt [3]{1+x^3}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 283, 337} \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{x^3+1}}{x}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3+1}\right )+\frac {\left (x^3+1\right )^{4/3}}{4 x^4} \]
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Rule 283
Rule 337
Rule 462
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^3\right )^{4/3}}{4 x^4}+\int \frac {\sqrt [3]{1+x^3}}{x^2} \, dx \\ & = -\frac {\sqrt [3]{1+x^3}}{x}+\frac {\left (1+x^3\right )^{4/3}}{4 x^4}+\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx \\ & = -\frac {\sqrt [3]{1+x^3}}{x}+\frac {\left (1+x^3\right )^{4/3}}{4 x^4}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=\frac {\left (1-3 x^3\right ) \sqrt [3]{1+x^3}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.30
method | result | size |
meijerg | \(\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}-\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}\) | \(30\) |
risch | \(-\frac {3 x^{6}+2 x^{3}-1}{4 x^{4} \left (x^{3}+1\right )^{\frac {2}{3}}}+\frac {x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}\) | \(42\) |
pseudoelliptic | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{4}+2 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}-4 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{4}-9 \left (x^{3}+1\right )^{\frac {1}{3}} x^{3}+3 \left (x^{3}+1\right )^{\frac {1}{3}}}{12 x^{4}}\) | \(107\) |
trager | \(-\frac {\left (3 x^{3}-1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{4 x^{4}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (218 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+337 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2358 x \left (x^{3}+1\right )^{\frac {2}{3}}-2358 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2140 x^{3}-218 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+1189 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1605\right )}{3}-\frac {\ln \left (218 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-773 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}-1803 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1585 x^{3}-218 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-753 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-634\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{3}+\frac {\ln \left (218 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-773 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}-1803 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1585 x^{3}-218 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-753 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-634\right )}{3}\) | \(417\) |
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Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {4 \, \sqrt {3} x^{4} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) + 2 \, x^{4} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (3 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{12 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=- \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{5}} \,d x } \]
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Time = 6.39 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^5} \, dx=\frac {{\left (x^3+1\right )}^{1/3}+x^3\,{\left (x^3+1\right )}^{1/3}}{4\,x^4}-\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -x^3\right )}{x} \]
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