Optimal. Leaf size=67 \[ \sqrt [3]{1-x^3}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 52, 59,
632, 210, 31} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [3]{1-x^3}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 52
Rule 59
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1-x^3}}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{1-x}}{x} \, dx,x,x^3\right )\\ &=\sqrt [3]{1-x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x)^{2/3} x} \, dx,x,x^3\right )\\ &=\sqrt [3]{1-x^3}-\frac {\log (x)}{2}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )\\ &=\sqrt [3]{1-x^3}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )\\ &=\sqrt [3]{1-x^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 = 0.05, size = 88, normalized size = 1.31 \begin {gather*} \sqrt [3]{1-x^3}-\frac {\tan ^{-1}\left (\frac {1+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]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.66, size = 49, normalized size = 0.73
method | result | size |
meijerg | \(-\frac {-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(49\) |
trager | \(\left (-x^{3}+1\right )^{\frac {1}{3}}+\frac {\ln \left (-\frac {211 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2704 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+8628 x^{3}+19749 \left (-x^{3}+1\right )^{\frac {2}{3}}-19749 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-1688 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-14247 \left (-x^{3}+1\right )^{\frac {1}{3}}+12559 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-7190}{x^{3}}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {1649 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+12981 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+10066 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}+14247 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-13192 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+19749 \left (-x^{3}+1\right )^{\frac {1}{3}}-32941 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-18694}{x^{3}}\right )}{3}\) | \(255\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.67, size = 71, normalized size = 1.06 \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 ) + {\left (-x^{3} + 1\right )}^{\frac {1}{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]
________________________________________________________________________________________
Fricas [A]
time = 0.98, size = 73, normalized size = 1.09 \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 ) + {\left (-x^{3} + 1\right )}^{\frac {1}{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]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 37, normalized size = 0.55 \begin {gather*} - \frac {x e^{\frac {i \pi }{3}} \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 {1}{x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.86, size = 72, normalized size = 1.07 \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 ) + {\left (-x^{3} + 1\right )}^{\frac {1}{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]
________________________________________________________________________________________
Mupad [B]
time = 0.37, size = 83, normalized size = 1.24 \begin {gather*} \frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left (3\,{\left (1-x^3\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 (1-x^3\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 )+{\left (1-x^3\right )}^{1/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________