Integrand size = 13, antiderivative size = 92 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{6 x^6}-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {1}{18} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {1}{36} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 58, 632, 210, 31} \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\left (x^6-1\right )^{2/3}}{6 x^6}-\frac {1}{12} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {\log (x)}{6} \]
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Rule 31
Rule 44
Rule 58
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^2} \, dx,x,x^6\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{6 x^6}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{6 x^6}+\frac {\log (x)}{6}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^6}\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{6 x^6}+\frac {\log (x)}{6}-\frac {1}{12} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^6}\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{6 x^6}-\frac {\arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\log (x)}{6}-\frac {1}{12} \log \left (1+\sqrt [3]{-1+x^6}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=\frac {1}{36} \left (\frac {6 \left (-1+x^6\right )^{2/3}}{x^6}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^6}\right )+\log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{6}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{36 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(96\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (-\frac {4 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], x^{6}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(97\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (2 \left (x^{6}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, x^{6}-2 \ln \left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right ) x^{6}+\ln \left (1-\left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right ) x^{6}+6 \left (x^{6}-1\right )^{\frac {2}{3}}}{36 \left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right ) \left (1-\left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )}\) | \(107\) |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{6}}+224 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) \ln \left (-\frac {24467220649221806672558383104 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )^{2} x^{6}-380817353737696452026579136 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) x^{6}-23171389162410581752275 x^{6}+275253794455679502932875392 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )-1565902121550195627043736518656 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )^{2}-275253794455679502932875392 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}+663622376189358973925865600 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )-142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+45974978496846392365625}{x^{6}}\right )+\frac {\ln \left (-\frac {24467220649221806672558383104 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )^{2} x^{6}+368680835558518968558048192 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) x^{6}-116115113033469041646202 x^{6}-275253794455679502932875392 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )-1565902121550195627043736518656 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )^{2}+275253794455679502932875392 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}+113114787277999968060114816 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}+114242288629703411942231}{x^{6}}\right )}{18}-224 \ln \left (-\frac {24467220649221806672558383104 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )^{2} x^{6}+368680835558518968558048192 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) x^{6}-116115113033469041646202 x^{6}-275253794455679502932875392 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )-1565902121550195627043736518656 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )^{2}+275253794455679502932875392 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}+113114787277999968060114816 \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}+114242288629703411942231}{x^{6}}\right ) \operatorname {RootOf}\left (16257024 \textit {\_Z}^{2}-4032 \textit {\_Z} +1\right )\) | \(441\) |
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Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{6} \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{6} \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + 6 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{36 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.35 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=- \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{8} \Gamma \left (\frac {7}{3}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{36} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{36} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{18} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 5.82 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \sqrt [3]{-1+x^6}} \, dx=\frac {{\left (x^6-1\right )}^{2/3}}{6\,x^6}-\ln \left (9\,{\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{36}\right )\,\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )+\ln \left (9\,{\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{36}\right )\,\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )-\frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{36}+\frac {1}{36}\right )}{18} \]
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