Optimal. Leaf size=84 \[ \frac {1}{2} \left (x^3+1\right )^{2/3}+\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {1}{6} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 0.76, 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, 50, 55, 618, 204, 31} \begin {gather*} \frac {1}{2} \left (x^3+1\right )^{2/3}+\frac {1}{2} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 50
Rule 55
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (1+x^3\right )^{2/3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (1+x^3\right )^{2/3}-\frac {\log (x)}{2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=\frac {1}{2} \left (1+x^3\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1+x^3}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=\frac {1}{2} \left (1+x^3\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 59, normalized size = 0.70 \begin {gather*} \frac {1}{2} \left (\left (x^3+1\right )^{2/3}+\log \left (1-\sqrt [3]{x^3+1}\right )-\log (x)\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.05, size = 84, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (1+x^3\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{6} \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 = 0.48, size = 65, normalized size = 0.77 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \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]
________________________________________________________________________________________
giac [A] time = 0.21, size = 64, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \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 = 2.38, size = 64, normalized size = 0.76
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi }\) | \(64\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2}+\frac {\ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-2 x^{3}-9 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9 \left (x^{3}+1\right )^{\frac {1}{3}}+19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-5}{x^{3}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+24 \left (x^{3}+1\right )^{\frac {1}{3}}+11 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+20}{x^{3}}\right )}{3}-\frac {\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+24 \left (x^{3}+1\right )^{\frac {1}{3}}+11 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+20}{x^{3}}\right )}{3}\) | \(357\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 63, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \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 = 0.82, size = 83, normalized size = 0.99 \begin {gather*} \frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}+\frac {{\left (x^3+1\right )}^{2/3}}{2}+\ln \left ({\left (x^3+1\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (x^3+1\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^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.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.84, size = 36, normalized size = 0.43 \begin {gather*} - \frac {x^{2} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________