Optimal. Leaf size=244 \[ -\frac {\tan ^{-1}\left (\frac {1+2 x^{2/3}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right ) \csc \left (\frac {2 \pi }{9}\right )\right ) \cos \left (\frac {\pi }{18}\right )-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{3} \tan ^{-1}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right ) \sin \left (\frac {\pi }{9}\right )-\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right ) \csc \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {2 \pi }{9}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {335, 281, 300,
648, 632, 210, 642, 31} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 x^{2/3}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{3} \cos \left (\frac {\pi }{18}\right ) \text {ArcTan}\left (\csc \left (\frac {2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right )\right )+\frac {1}{3} \sin \left (\frac {\pi }{9}\right ) \text {ArcTan}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right )-\frac {1}{3} \sin \left (\frac {2 \pi }{9}\right ) \text {ArcTan}\left (\csc \left (\frac {\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right )-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )+1\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )+1\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )+1\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 31
Rule 210
Rule 281
Rule 300
Rule 335
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x}}{1-x^6} \, dx &=3 \text {Subst}\left (\int \frac {x^3}{1-x^{18}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{2} \text {Subst}\left (\int \frac {x}{1-x^9} \, dx,x,x^{2/3}\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {1}{2}-\frac {x}{2}}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {\cos \left (\frac {\pi }{9}\right )+x \cos \left (\frac {2 \pi }{9}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {-x \cos \left (\frac {\pi }{9}\right )-\sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {-\cos \left (\frac {2 \pi }{9}\right )+x \sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )\\ &=-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^{2/3}\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \text {Subst}\left (\int \frac {2 x-2 \sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \text {Subst}\left (\int \frac {2 x+2 \cos \left (\frac {\pi }{9}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac {1}{3} \left (\cos \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \text {Subst}\left (\int \frac {2 x-2 \cos \left (\frac {2 \pi }{9}\right )}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac {1}{3} \left (\left (1+\cos \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \left (\sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )\\ &=-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^{2/3}\right )-\frac {1}{3} \left (2 \cos \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {2 \pi }{9}\right )} \, dx,x,2 x^{2/3}-2 \cos \left (\frac {2 \pi }{9}\right )\right )-\frac {1}{3} \left (2 \left (1+\cos \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {\pi }{18}\right )} \, dx,x,2 x^{2/3}-2 \sin \left (\frac {\pi }{18}\right )\right )+\frac {1}{3} \left (2 \sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {\pi }{9}\right )} \, dx,x,2 \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1+2 x^{2/3}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )+\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right ) \csc \left (\frac {2 \pi }{9}\right )\right ) \cot \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{3} \tan ^{-1}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right ) \sin \left (\frac {\pi }{9}\right )-\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right ) \csc \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {2 \pi }{9}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.14, size = 219, normalized size = 0.90 \begin {gather*} \frac {1}{12} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x}}{\sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{x}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{x}\right )-2 \log \left (1+\sqrt [3]{x}\right )+\log \left (1-\sqrt [3]{x}+x^{2/3}\right )+\log \left (1+\sqrt [3]{x}+x^{2/3}\right )+2 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]+2 \text {RootSum}\left [1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 10.57, size = 162, normalized size = 0.66
method | result | size |
derivativedivides | \(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}-\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(162\) |
default | \(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}-\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(162\) |
meijerg | \(-\frac {x^{\frac {4}{3}} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{9}}\right )+\cos \left (\frac {4 \pi }{9}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )-2 \sin \left (\frac {4 \pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1-\cos \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )-\cos \left (\frac {\pi }{9}\right ) \ln \left (1-2 \cos \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )-2 \sin \left (\frac {\pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1-\cos \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{9}}}{2+\left (x^{6}\right )^{\frac {1}{9}}}\right )+\cos \left (\frac {2 \pi }{9}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )+2 \sin \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1+\cos \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {2}{9}}}\) | \(231\) |
trager | \(\text {Expression too large to display}\) | \(1949\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 555 vs.
\(2 (167) = 334\).
time = 0.48, size = 555, normalized size = 2.27 \begin {gather*} \frac {1}{3} \, {\left (\sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) - \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (-\frac {8 \, {\left (2 \, \cos \left (\frac {1}{9} \, \pi \right )^{3} - \cos \left (\frac {1}{9} \, \pi \right )\right )} \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )} x^{\frac {2}{3}} - 2 \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )} \sqrt {{\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} - 1\right )} x^{\frac {2}{3}} + x^{\frac {4}{3}} + 1} + \sqrt {3}}{16 \, \cos \left (\frac {1}{9} \, \pi \right )^{4} - 16 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} + 3}\right ) - \frac {1}{3} \, {\left (\sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) + \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (\frac {16 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} - 8 \, x^{\frac {2}{3}} + \sqrt {-128 \, {\left (2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} - 1\right )} x^{\frac {2}{3}} + 64 \, x^{\frac {4}{3}} + 64} - 8}{16 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )}\right ) - \frac {1}{12} \, {\left (\sqrt {3} \sin \left (\frac {1}{9} \, \pi \right ) + \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left (64 \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} - 1\right )} x^{\frac {2}{3}} + 64 \, x^{\frac {4}{3}} + 64\right ) + \frac {1}{6} \, \cos \left (\frac {1}{9} \, \pi \right ) \log \left (-16 \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - 2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} + 1\right )} x^{\frac {2}{3}} + 16 \, x^{\frac {4}{3}} + 16\right ) + \frac {1}{12} \, {\left (\sqrt {3} \sin \left (\frac {1}{9} \, \pi \right ) - \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left (-128 \, {\left (2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} - 1\right )} x^{\frac {2}{3}} + 64 \, x^{\frac {4}{3}} + 64\right ) + \frac {2}{3} \, \arctan \left (-\frac {16 \, {\left (2 \, \cos \left (\frac {1}{9} \, \pi \right )^{3} - \cos \left (\frac {1}{9} \, \pi \right )\right )} \sin \left (\frac {1}{9} \, \pi \right ) - 4 \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )} x^{\frac {2}{3}} + {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )} \sqrt {-16 \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - 2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} + 1\right )} x^{\frac {2}{3}} + 16 \, x^{\frac {4}{3}} + 16} - 2 \, \sqrt {3}}{2 \, {\left (16 \, \cos \left (\frac {1}{9} \, \pi \right )^{4} - 16 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} + 3\right )}}\right ) \sin \left (\frac {1}{9} \, \pi \right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left (x^{\frac {2}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.54, size = 217, normalized size = 0.89 \begin {gather*} \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {4}{9} \, \pi \right )}{\sin \left (\frac {4}{9} \, \pi \right )}\right ) \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) + \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {2}{9} \, \pi \right )}{\sin \left (\frac {2}{9} \, \pi \right )}\right ) \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) - \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} + \cos \left (\frac {1}{9} \, \pi \right )}{\sin \left (\frac {1}{9} \, \pi \right )}\right ) \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \frac {1}{6} \, {\left (\cos \left (\frac {4}{9} \, \pi \right )^{2} - \sin \left (\frac {4}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {4}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {2}{9} \, \pi \right )^{2} - \sin \left (\frac {2}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {2}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {1}{9} \, \pi \right )^{2} - \sin \left (\frac {1}{9} \, \pi \right )^{2}\right )} \log \left (2 \, x^{\frac {2}{3}} \cos \left (\frac {1}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | x^{\frac {2}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 186, normalized size = 0.76 \begin {gather*} -\frac {\ln \left (43046721\,x^{2/3}-43046721\right )}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}}{6}-\ln \left (55788550416\,x^{2/3}\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (55788550416\,x^{2/3}\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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