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 = 84, normalized size of antiderivative = 0.87, 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, 60, 632, 210, 31} \[ \int \frac {\sqrt [3]{-1+x^3}}{x^7} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\sqrt [3]{x^3-1}}{18 x^3}+\frac {1}{18} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {\sqrt [3]{x^3-1}}{6 x^6}-\frac {\log (x)}{18} \]
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Rule 31
Rule 43
Rule 44
Rule 60
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}{(-1+x)^{2/3} x^2} \, 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}{(-1+x)^{2/3} x} \, 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 = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^7} \, dx=\frac {1}{54} \left (\frac {3 \left (-3+x^3\right ) \sqrt [3]{-1+x^3}}{x^6}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [3]{-1+x^3}\right )-\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.43 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(81\) |
risch | \(\frac {x^{6}-4 x^{3}+3}{18 x^{6} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(89\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{6}+2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) x^{6}-\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) x^{6}+3 x^{3} \left (x^{3}-1\right )^{\frac {1}{3}}-9 \left (x^{3}-1\right )^{\frac {1}{3}}}{54 {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{2} {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{2}}\) | \(120\) |
trager | \(\frac {\left (x^{3}-3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{18 x^{6}}-\frac {\ln \left (-\frac {5890048 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}+630720 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}-352128 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-47120384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+352128 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-1088384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{27}-\frac {64 \ln \left (-\frac {5890048 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}+630720 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}-352128 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-47120384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+352128 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-1088384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right ) \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )}{27}+\frac {64 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \ln \left (-\frac {5890048 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}-446656 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+352128 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-47120384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}-352128 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-384128 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{27}\) | \(446\) |
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Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^7} \, dx=\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x^{3} - 3\right )}}{54 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^7} \, dx=- \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} \]
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Time = 0.27 (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 = 79, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^7} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} - \frac {1}{54} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 6.41 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^7} \, dx=\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{81}+\frac {1}{81}\right )}{27}-\frac {\frac {{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{18}}{{\left (x^3-1\right )}^2+2\,x^3-1}-\ln \left (\frac {1}{6}-\frac {{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \]
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