Integrand size = 16, antiderivative size = 105 \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (15-18 x-20 x^3+27 x^4\right )}{90 x^6}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2}{27} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{27} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1858, 272, 44, 57, 632, 210, 31, 277, 270} \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=-\frac {2 \arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2 \left (x^3+1\right )^{2/3}}{9 x^3}-\frac {1}{9} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\left (x^3+1\right )^{2/3}}{6 x^6}-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9} \]
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Rule 31
Rule 44
Rule 57
Rule 210
Rule 270
Rule 272
Rule 277
Rule 632
Rule 1858
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^7 \sqrt [3]{1+x^3}}+\frac {1}{x^6 \sqrt [3]{1+x^3}}\right ) \, dx \\ & = -\int \frac {1}{x^7 \sqrt [3]{1+x^3}} \, dx+\int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^3\right ) \\ & = \frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2}{27} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right ) \\ & = \frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )-\frac {1}{9} \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}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^3}\right )+\frac {2}{9} \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}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\log (x)}{9}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.39 \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-2+3 x^3+5 x^5 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},3,\frac {5}{3},1+x^3\right )\right )}{10 x^5} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.69 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {27 x^{7}-20 x^{6}+9 x^{4}-5 x^{3}-18 x +15}{90 x^{6} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{27 \pi }\) | \(101\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {28 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {10}{3}\right ], \left [2, 4\right ], -x^{3}\right )}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \left (\frac {9}{4}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{6}}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (1-\frac {3 x^{3}}{2}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{5 x^{5}}\) | \(110\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (27 x^{4}-20 x^{3}-18 x +15\right )}{90 x^{6}}+\frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}+18 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}+9 \left (x^{3}+1\right )^{\frac {2}{3}}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}+9 \left (x^{3}+1\right )^{\frac {1}{3}}+38 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+5}{x^{3}}\right )}{27}+\frac {2 \ln \left (\frac {-16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}+34 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-15 x^{3}-24 \left (x^{3}+1\right )^{\frac {2}{3}}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-24 \left (x^{3}+1\right )^{\frac {1}{3}}+22 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-20}{x^{3}}\right )}{27}-\frac {4 \ln \left (\frac {-16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}+34 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-15 x^{3}-24 \left (x^{3}+1\right )^{\frac {2}{3}}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-24 \left (x^{3}+1\right )^{\frac {1}{3}}+22 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-20}{x^{3}}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{27}\) | \(454\) |
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Time = 0.49 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=\frac {20 \, \sqrt {3} x^{6} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - 10 \, x^{6} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (27 \, x^{4} - 20 \, x^{3} - 18 \, x + 15\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 2.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=\frac {\left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 x^{5} \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{7} \Gamma \left (\frac {10}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.11 \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=-\frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {4 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{18 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {1}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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\[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{7}} \,d x } \]
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Time = 6.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.39 \[ \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx=-\frac {2\,\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-\frac {4}{81}\right )}{27}-\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-9\,{\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )}^2\right )\,\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-9\,{\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )}^2\right )\,\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )-\frac {2\,{\left (x^3+1\right )}^{2/3}-3\,x^3\,{\left (x^3+1\right )}^{2/3}}{10\,x^5}-\frac {\frac {7\,{\left (x^3+1\right )}^{2/3}}{18}-\frac {2\,{\left (x^3+1\right )}^{5/3}}{9}}{2\,x^3-{\left (x^3+1\right )}^2+1} \]
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