Integrand size = 18, antiderivative size = 99 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {\left (-3-5 x^3\right ) \sqrt [3]{-1+x^3}}{18 x^6}-\frac {4 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {4}{27} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {2}{27} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {457, 79, 43, 60, 632, 210, 31} \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=-\frac {4 \arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {4 \sqrt [3]{x^3-1}}{9 x^3}+\frac {2}{9} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {\left (x^3-1\right )^{4/3}}{6 x^6}-\frac {2 \log (x)}{9} \]
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Rule 31
Rule 43
Rule 60
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x} (1+x)}{x^3} \, dx,x,x^3\right ) \\ & = \frac {\left (-1+x^3\right )^{4/3}}{6 x^6}+\frac {4}{9} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}+\frac {4}{27} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right ) \\ & = -\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {2 \log (x)}{9}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {2}{9} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ & = -\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {2 \log (x)}{9}+\frac {2}{9} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {4}{9} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right ) \\ & = -\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {4 \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2 \log (x)}{9}+\frac {2}{9} \log \left (1+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {1}{54} \left (-\frac {3 \sqrt [3]{-1+x^3} \left (3+5 x^3\right )}{x^6}-8 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+8 \log \left (1+\sqrt [3]{-1+x^3}\right )-4 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.43 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {5 x^{6}-2 x^{3}-3}{18 x^{6} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(91\) |
pseudoelliptic | \(\frac {8 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{6}+8 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) x^{6}-4 \ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) x^{6}-15 x^{3} \left (x^{3}-1\right )^{\frac {1}{3}}-9 \left (x^{3}-1\right )^{\frac {1}{3}}}{54 {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{2} {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{2}}\) | \(120\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(156\) |
trager | \(-\frac {\left (5 x^{3}+3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{18 x^{6}}+\frac {256 \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}-\frac {4 \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}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-352128 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+47120384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+1088384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-2743}{x^{3}}\right )}{27}-\frac {256 \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}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-352128 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+47120384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+1088384 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-2743}{x^{3}}\right ) \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )}{27}\) | \(446\) |
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Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {8 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 4 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 8 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (5 \, x^{3} + 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 87.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=- \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} - \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {4}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {2}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {4}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {4}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} + 8 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} - \frac {2}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {4}{27} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 6.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{9}+\frac {1}{9}\right )}{9}+\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{81}+\frac {1}{81}\right )}{27}-\frac {\frac {{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{18}}{{\left (x^3-1\right )}^2+2\,x^3-1}+\ln \left ({\left (x^3-1\right )}^{1/3}-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^3-1\right )}^{1/3}}{3\,x^3}-\ln \left (\frac {1}{2}-{\left (x^3-1\right )}^{1/3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {1}{6}-\frac {{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \]
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