Optimal. Leaf size=90 \[ -\frac {\sqrt [3]{x^3+1}}{3 x^3}+\frac {1}{9} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {1}{18} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 47, 57, 618, 204, 31} \begin {gather*} -\frac {\sqrt [3]{x^3+1}}{3 x^3}+\frac {1}{6} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 47
Rule 57
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1+x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{3 x^3}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{3 x^3}-\frac {\log (x)}{6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{3 x^3}-\frac {\log (x)}{6}+\frac {1}{6} \log \left (1-\sqrt [3]{1+x^3}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{3 x^3}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{6}+\frac {1}{6} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 26, normalized size = 0.29 \begin {gather*} \frac {1}{4} \left (x^3+1\right )^{4/3} \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.08, size = 90, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [3]{1+x^3}}{3 x^3}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{18} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 78, normalized size = 0.87 \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 ) + 6 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{18 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 67, normalized size = 0.74 \begin {gather*} -\frac {1}{9} \, \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 {1}{3}}}{3 \, x^{3}} - \frac {1}{18} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{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.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 2.23, size = 55, normalized size = 0.61
method | result | size |
meijerg | \(-\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(55\) |
risch | \(-\frac {\left (x^{3}+1\right )^{\frac {1}{3}}}{3 x^{3}}+\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(59\) |
trager | \(-\frac {\left (x^{3}+1\right )^{\frac {1}{3}}}{3 x^{3}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+24 \left (x^{3}+1\right )^{\frac {1}{3}}+11 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+20}{x^{3}}\right )}{9}-\frac {\ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-2 x^{3}-9 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9 \left (x^{3}+1\right )^{\frac {1}{3}}+19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-5}{x^{3}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{9}-\frac {\ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-2 x^{3}-9 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9 \left (x^{3}+1\right )^{\frac {1}{3}}+19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-5}{x^{3}}\right )}{9}\) | \(359\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 66, normalized size = 0.73 \begin {gather*} -\frac {1}{9} \, \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 {1}{3}}}{3 \, x^{3}} - \frac {1}{18} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{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.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.90, size = 78, normalized size = 0.87 \begin {gather*} \frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.88, size = 32, normalized size = 0.36 \begin {gather*} - \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.
[In]
[Out]
________________________________________________________________________________________