Optimal. Leaf size=99 \[ \frac {\left (3-13 x^3\right ) \sqrt [3]{-1+x^3}}{18 x^6}-\frac {5 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {5}{27} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{54} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 84, normalized size of antiderivative = 0.85, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {457, 79, 43, 60,
632, 210, 31} \begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {5 \sqrt [3]{x^3-1}}{9 x^3}+\frac {5}{18} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {\left (x^3-1\right )^{4/3}}{6 x^6}-\frac {5 \log (x)}{18} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 60
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^7} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x} (-1+2 x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}+\frac {5}{9} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{9 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}+\frac {5}{27} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{9 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {5 \log (x)}{18}+\frac {5}{18} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {5}{18} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{9 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {5 \log (x)}{18}+\frac {5}{18} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{9} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{9 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {5 \log (x)}{18}+\frac {5}{18} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 92, normalized size = 0.93 \begin {gather*} \frac {1}{54} \left (\frac {3 \left (3-13 x^3\right ) \sqrt [3]{-1+x^3}}{x^6}-10 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+10 \log \left (1+\sqrt [3]{-1+x^3}\right )-5 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.79, size = 91, normalized size = 0.92
method | result | size |
risch | \(-\frac {13 x^{6}-16 x^{3}+3}{18 x^{6} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(91\) |
meijerg | \(\frac {2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}+\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\) | \(156\) |
trager | \(-\frac {\left (13 x^{3}-3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{18 x^{6}}+\frac {5 \ln \left (\frac {-864256 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}-200064 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+352128 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-11543 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}+1263936 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+6914048 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+5502 \left (x^{3}-1\right )^{\frac {1}{3}}+1019840 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+21437}{x^{3}}\right )}{27}-\frac {5 \ln \left (-\frac {-6754304 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}+619712 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+352128 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1266 x^{3}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-911808 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+54034432 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+5502 \left (x^{3}-1\right )^{\frac {1}{3}}-419648 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-1055}{x^{3}}\right )}{27}-\frac {320 \ln \left (-\frac {-6754304 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}+619712 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+352128 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1266 x^{3}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-911808 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+54034432 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+5502 \left (x^{3}-1\right )^{\frac {1}{3}}-419648 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-1055}{x^{3}}\right ) \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )}{27}\) | \(436\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 103, normalized size = 1.04 \begin {gather*} \frac {5}{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 {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {5}{54} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 88, normalized size = 0.89 \begin {gather*} \frac {10 \, \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 ) - 5 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (13 \, x^{3} - 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 114.60, size = 68, normalized size = 0.69 \begin {gather*} - \frac {2 \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 81, normalized size = 0.82 \begin {gather*} \frac {5}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} + 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} - \frac {5}{54} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{27} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 187, normalized size = 1.89 \begin {gather*} \frac {2\,\ln \left (\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}+\frac {4}{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}-\frac {2\,{\left (x^3-1\right )}^{1/3}}{3\,x^3}-\ln \left (1-2\,{\left (x^3-1\right )}^{1/3}+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (2\,{\left (x^3-1\right )}^{1/3}-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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