Optimal. Leaf size=55 \[ \frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 57, 632,
210, 31} \begin {gather*} \frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{1-x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right )\\ &=-\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 )\\ &=-\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 )\\ &=\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 = 0.04, size = 79, normalized size = 1.44 \begin {gather*} \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]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.02, size = 16, normalized size = 0.29 \begin {gather*} -\frac {\text {hyper}\left [\left \{\frac {1}{3},\frac {1}{3}\right \},\left \{\frac {4}{3}\right \},\frac {1}{x^3}\right ]}{x} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.93, size = 65, normalized size = 1.18
method | result | size |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi }\) | \(65\) |
trager | \(\frac {\ln \left (\frac {-211 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3126 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-11543 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}-19749 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+1688 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5502 \left (-x^{3}+1\right )^{\frac {1}{3}}+15935 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+21437}{x^{3}}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {13 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-70 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+39 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-48 x^{3}-105 \left (-x^{3}+1\right )^{\frac {2}{3}}-144 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-104 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-39 \left (-x^{3}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+40}{x^{3}}\right )}{3}\) | \(243\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 62, normalized size = 1.13 \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}{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 = 0.31, size = 64, normalized size = 1.16 \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}{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 = 0.42, size = 32, normalized size = 0.58 \begin {gather*} - \frac {e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {1}{x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 77, normalized size = 1.40 \begin {gather*} -\frac {\ln \left (\left (\left (-x^{3}+1\right )^{\frac {1}{3}}\right )^{2}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right )}{6}+\frac {1}{3} \sqrt {3} \arctan \left (\frac {2 \left (\left (-x^{3}+1\right )^{\frac {1}{3}}+\frac 1{2}\right )}{\sqrt {3}}\right )+\frac {\ln \left |\left (-x^{3}+1\right )^{\frac {1}{3}}-1\right |}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.51, size = 80, normalized size = 1.45 \begin {gather*} \frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (1-x^3\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 (1-x^3\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]
________________________________________________________________________________________