Optimal. Leaf size=97 \[ \frac {\sqrt [3]{x^3+1} \left (3 x^3+1\right )}{3 x^3}+\frac {2}{9} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {1}{9} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {446, 78, 50, 57, 618, 204, 31} \begin {gather*} \frac {\left (x^3+1\right )^{4/3}}{3 x^3}+\frac {2}{3} \sqrt [3]{x^3+1}+\frac {1}{3} \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 )}{3 \sqrt {3}}-\frac {\log (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 78
Rule 204
Rule 446
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-1+x) \sqrt [3]{1+x}}{x^2} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{4/3}}{3 x^3}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}-\frac {\log (x)}{3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=\frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}-\frac {\log (x)}{3}+\frac {1}{3} \log \left (1-\sqrt [3]{1+x^3}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=\frac {2}{3} \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{3 x^3}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{3} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 98, normalized size = 1.01 \begin {gather*} \frac {\sqrt [3]{x^3+1}}{3 x^3}+\sqrt [3]{x^3+1}+\frac {2}{9} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {1}{9} \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}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 97, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+x^3} \left (1+3 x^3\right )}{3 x^3}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{9} \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 = 0.53, size = 85, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (3 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 74, normalized size = 0.76 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \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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maple [C] time = 2.19, size = 71, normalized size = 0.73
method | result | size |
risch | \(\frac {3 x^{6}+4 x^{3}+1}{3 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}}+\frac {-\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{27}+\frac {2 \left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{9}}{\Gamma \left (\frac {2}{3}\right )}\) | \(71\) |
meijerg | \(\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {-\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(103\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (3 x^{3}+1\right )}{3 x^{3}}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+11 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{9}-\frac {2 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{9}-\frac {2 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{9}\) | \(368\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 73, normalized size = 0.75 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \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 = 1.24, size = 152, normalized size = 1.57 \begin {gather*} \frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+{\left (x^3+1\right )}^{1/3}-\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 )+\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 (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 87.87, size = 65, normalized size = 0.67 \begin {gather*} - \frac {x \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} + \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^{i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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