Optimal. Leaf size=105 \[ -\frac {2}{27} \log \left (\sqrt [3]{x^3+1}-1\right )+\frac {1}{27} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (27 x^4-20 x^3-18 x+15\right )}{90 x^6} \]
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Rubi [A] time = 0.10, antiderivative size = 118, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1844, 266, 51, 55, 618, 204, 31, 271, 264} \begin {gather*} -\frac {2 \left (x^3+1\right )^{2/3}}{9 x^3}-\frac {1}{9} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3}}{6 x^6}-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 264
Rule 266
Rule 271
Rule 618
Rule 1844
Rubi steps
\begin {align*} \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx &=\int \left (-\frac {1}{x^7 \sqrt [3]{1+x^3}}+\frac {1}{x^6 \sqrt [3]{1+x^3}}\right ) \, dx\\ &=-\int \frac {1}{x^7 \sqrt [3]{1+x^3}} \, dx+\int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2}{27} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^3}\right )+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\log (x)}{9}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 41, normalized size = 0.39 \begin {gather*} \frac {\left (x^3+1\right )^{2/3} \left (5 x^5 \, _2F_1\left (\frac {2}{3},3;\frac {5}{3};x^3+1\right )+3 x^3-2\right )}{10 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 15.71, size = 105, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (15-18 x-20 x^3+27 x^4\right )}{90 x^6}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2}{27} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{27} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 114, normalized size = 1.09 \begin {gather*} \frac {20 \, \sqrt {3} x^{6} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - 10 \, x^{6} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (27 \, x^{4} - 20 \, x^{3} - 18 \, x + 15\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.76, size = 101, normalized size = 0.96
method | result | size |
risch | \(\frac {27 x^{7}-20 x^{6}+9 x^{4}-5 x^{3}-18 x +15}{90 x^{6} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{27 \pi }\) | \(101\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {28 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {10}{3}\right ], \left [2, 4\right ], -x^{3}\right )}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \left (\frac {9}{4}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{6}}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (1-\frac {3 x^{3}}{2}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{5 x^{5}}\) | \(110\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (27 x^{4}-20 x^{3}-18 x +15\right )}{90 x^{6}}+\frac {4 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}+18 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}+30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}+30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}+38 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{27}+\frac {2 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}-34 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}-22 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{27}-\frac {4 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}-34 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}-22 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{27}\) | \(456\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 117, normalized size = 1.11 \begin {gather*} -\frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {4 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{18 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {1}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{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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mupad [B] time = 1.15, size = 146, normalized size = 1.39 \begin {gather*} -\frac {2\,\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-\frac {4}{81}\right )}{27}-\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-9\,{\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )}^2\right )\,\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-9\,{\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )}^2\right )\,\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )-\frac {2\,{\left (x^3+1\right )}^{2/3}-3\,x^3\,{\left (x^3+1\right )}^{2/3}}{10\,x^5}-\frac {\frac {7\,{\left (x^3+1\right )}^{2/3}}{18}-\frac {2\,{\left (x^3+1\right )}^{5/3}}{9}}{2\,x^3-{\left (x^3+1\right )}^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.16, size = 82, normalized size = 0.78 \begin {gather*} \frac {\left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 x^{5} \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{7} \Gamma \left (\frac {10}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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