Integrand size = 13, antiderivative size = 97 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {\left (-3-x^3\right ) \sqrt [3]{1+x^3}}{18 x^6}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{27} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{54} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {272, 43, 44, 59, 632, 210, 31} \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\sqrt [3]{x^3+1}}{18 x^3}-\frac {1}{18} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {\sqrt [3]{x^3+1}}{6 x^6}+\frac {\log (x)}{18} \]
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Rule 31
Rule 43
Rule 44
Rule 59
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{1+x^3}}{6 x^6}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}-\frac {1}{27} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}+\frac {\log (x)}{18}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {1}{18} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right ) \\ & = -\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}+\frac {\log (x)}{18}-\frac {1}{18} \log \left (1-\sqrt [3]{1+x^3}\right )-\frac {1}{9} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right ) \\ & = -\frac {\sqrt [3]{1+x^3}}{6 x^6}-\frac {\sqrt [3]{1+x^3}}{18 x^3}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\log (x)}{18}-\frac {1}{18} \log \left (1-\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {1}{54} \left (-\frac {3 \sqrt [3]{1+x^3} \left (3+x^3\right )}{x^6}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{1+x^3}\right )+\log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.71 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62
method | result | size |
meijerg | \(-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(60\) |
risch | \(-\frac {x^{6}+4 x^{3}+3}{18 x^{6} \left (x^{3}+1\right )^{\frac {2}{3}}}-\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27 \Gamma \left (\frac {2}{3}\right )}\) | \(69\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{6}-2 \ln \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right ) x^{6}+\ln \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right ) x^{6}-3 x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}-9 \left (x^{3}+1\right )^{\frac {1}{3}}}{54 {\left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right )}^{2} {\left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )}^{2}}\) | \(115\) |
trager | \(-\frac {\left (x^{3}+3\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{18 x^{6}}-\frac {\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-12 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+24 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {2}{3}}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-9 \left (x^{3}+1\right )^{\frac {1}{3}}-16}{x^{3}}\right )}{27}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-8 x^{3}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-9 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-9 \left (x^{3}+1\right )^{\frac {2}{3}}+29 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}-20}{x^{3}}\right )}{27}\) | \(271\) |
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Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{3} + 3\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=- \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{54} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} + \frac {1}{54} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 6.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{1+x^3}}{x^7} \, dx=\frac {\frac {{\left (x^3+1\right )}^{1/3}}{9}+\frac {{\left (x^3+1\right )}^{4/3}}{18}}{2\,x^3-{\left (x^3+1\right )}^2+1}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{81}-\frac {1}{81}\right )}{27}-\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{3}+\frac {1}{6}-\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{3}+\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \]
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