Optimal. Leaf size=137 \[ \frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {446, 86, 57, 618, 204, 31, 617} \begin {gather*} \frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 57
Rule 86
Rule 204
Rule 446
Rule 617
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} x (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} x} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\log (x)}{2}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\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 {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {\log (x)}{2}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2^{2/3}}+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 179, normalized size = 1.31 \begin {gather*} \frac {1}{12} \left (4 \log \left (1-\sqrt [3]{1-x^3}\right )-2 \sqrt [3]{2} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )+\sqrt [3]{2} \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{2-2 x^3}+2^{2/3}\right )-2 \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{1-x^3}+1\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )+2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.19, size = 195, normalized size = 1.42 \begin {gather*} \frac {1}{3} \log \left (\sqrt [3]{1-x^3}-1\right )-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}-2\right )}{3\ 2^{2/3}}-\frac {1}{6} \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{1-x^3}+1\right )+\frac {\log \left (\sqrt [3]{2} \left (1-x^3\right )^{2/3}+2^{2/3} \sqrt [3]{1-x^3}+2\right )}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 182, normalized size = 1.33 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \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}{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 = 0.24, size = 149, normalized size = 1.09 \begin {gather*} \frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \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}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \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 [F] time = 2.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+1\right ) x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.92, size = 344, normalized size = 2.51 \begin {gather*} \frac {\ln \left (5-5\,{\left (1-x^3\right )}^{1/3}\right )}{3}-\frac {2^{1/3}\,\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,\left (\frac {2^{2/3}\,\left (243\,2^{1/3}+243\,{\left (1-x^3\right )}^{1/3}\right )}{36}+9\right )}{6}\right )}{6}+\ln \left (\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left ({\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\,\left (243\,{\left (1-x^3\right )}^{1/3}+243-\sqrt {3}\,243{}\mathrm {i}\right )+9\right )+6\,{\left (1-x^3\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left ({\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\,\left (243\,{\left (1-x^3\right )}^{1/3}+243+\sqrt {3}\,243{}\mathrm {i}\right )+9\right )\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (\frac {{\left (-1\right )}^{2/3}\,2^{2/3}\,\left (243\,{\left (-1\right )}^{1/3}\,2^{1/3}-243\,{\left (1-x^3\right )}^{1/3}\right )}{36}-9\right )}{6}\right )}{6}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (243\,{\left (1-x^3\right )}^{1/3}+\frac {243\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )}{144}+9\right )}{12}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________