Optimal. Leaf size=418 \[ -\frac {1}{2 x^2}+\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \]
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Rubi [A]
time = 0.23, antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1518, 12,
1388, 206, 31, 648, 631, 210, 642} \begin {gather*} \frac {\left (\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (-\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {1}{2 x^2}+\frac {\left (3+i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1388
Rule 1518
Rubi steps
\begin {align*} \int \frac {1-x^3}{x^3 \left (1-x^3+x^6\right )} \, dx &=-\frac {1}{2 x^2}-\frac {1}{2} \int \frac {2 x^3}{1-x^3+x^6} \, dx\\ &=-\frac {1}{2 x^2}-\int \frac {x^3}{1-x^3+x^6} \, dx\\ &=-\frac {1}{2 x^2}+\frac {1}{6} \left (-3+i \sqrt {3}\right ) \int \frac {1}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^3} \, dx-\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^3} \, dx\\ &=-\frac {1}{2 x^2}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {-2^{2/3} \sqrt [3]{1+i \sqrt {3}}-x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {-2^{2/3} \sqrt [3]{1-i \sqrt {3}}-x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}\\ &=-\frac {1}{2 x^2}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\\ &=-\frac {1}{2 x^2}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}\\ &=-\frac {1}{2 x^2}+\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 47, normalized size = 0.11 \begin {gather*} -\frac {1}{2 x^2}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\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 = 0.04, size = 46, normalized size = 0.11
method | result | size |
risch | \(-\frac {1}{2 x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (-18 \textit {\_R}^{4}+3 \textit {\_R} +x \right )\right )}{3}\) | \(38\) |
default | \(-\frac {1}{2 x^{2}}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(46\) |
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. 802 vs.
\(2 (272) = 544\).
time = 0.42, size = 802, normalized size = 1.92 \begin {gather*} \frac {2 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} x^{2} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) \log \left (-36 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 324 \, x^{2} + 54 \cdot 18^{\frac {1}{3}} 12^{\frac {1}{3}}\right ) - 8 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} x^{2} \arctan \left (-\frac {18 \cdot 18^{\frac {1}{3}} 12^{\frac {5}{6}} \sqrt {3} x - 18^{\frac {1}{3}} 12^{\frac {5}{6}} \sqrt {3} \sqrt {-36 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 324 \, x^{2} + 54 \cdot 18^{\frac {1}{3}} 12^{\frac {1}{3}}} - 648 \, \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )}{648 \, \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )}\right ) \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 4 \, {\left (18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x^{2} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 18^{\frac {2}{3}} 12^{\frac {1}{6}} x^{2} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} \arctan \left (-\frac {6 \cdot 18^{\frac {1}{3}} 12^{\frac {5}{6}} \sqrt {3} x \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - \sqrt {2} \sqrt {18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 3 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} x \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 18 \, x^{2} + 3 \cdot 18^{\frac {1}{3}} 12^{\frac {1}{3}}} {\left (18^{\frac {1}{3}} 12^{\frac {5}{6}} \sqrt {3} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 3 \cdot 18^{\frac {1}{3}} 12^{\frac {5}{6}} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} + 18 \, {\left (18^{\frac {1}{3}} 12^{\frac {5}{6}} x + 24 \, \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 108 \, \sqrt {3}}{108 \, {\left (4 \, \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )^{2} - 3\right )}}\right ) - 4 \, {\left (18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x^{2} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 18^{\frac {2}{3}} 12^{\frac {1}{6}} x^{2} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} \arctan \left (\frac {6 \cdot 18^{\frac {1}{3}} 12^{\frac {5}{6}} \sqrt {3} x \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - \sqrt {2} \sqrt {18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 3 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} x \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 18 \, x^{2} + 3 \cdot 18^{\frac {1}{3}} 12^{\frac {1}{3}}} {\left (18^{\frac {1}{3}} 12^{\frac {5}{6}} \sqrt {3} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 3 \cdot 18^{\frac {1}{3}} 12^{\frac {5}{6}} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} - 18 \, {\left (18^{\frac {1}{3}} 12^{\frac {5}{6}} x - 24 \, \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 108 \, \sqrt {3}}{108 \, {\left (4 \, \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )^{2} - 3\right )}}\right ) + {\left (18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x^{2} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 18^{\frac {2}{3}} 12^{\frac {1}{6}} x^{2} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} \log \left (72 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 216 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} x \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 1296 \, x^{2} + 216 \cdot 18^{\frac {1}{3}} 12^{\frac {1}{3}}\right ) - {\left (18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x^{2} \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 18^{\frac {2}{3}} 12^{\frac {1}{6}} x^{2} \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right )\right )} \log \left (72 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} \sqrt {3} x \sin \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) - 216 \cdot 18^{\frac {2}{3}} 12^{\frac {1}{6}} x \cos \left (\frac {2}{3} \, \arctan \left (\sqrt {3} - 2\right )\right ) + 1296 \, x^{2} + 216 \cdot 18^{\frac {1}{3}} 12^{\frac {1}{3}}\right ) - 54}{108 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 32, normalized size = 0.08 \begin {gather*} - \operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log {\left (- 1458 t^{4} - 9 t + x \right )} \right )\right )} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 645 vs. \(2 (272) = 544\).
time = 4.19, size = 645, normalized size = 1.54 \begin {gather*} \frac {1}{9} \, {\left (2 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{4} - 12 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{2} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 2 \, \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right )^{4} + 8 \, \cos \left (\frac {4}{9} \, \pi \right )^{3} \sin \left (\frac {4}{9} \, \pi \right ) - 8 \, \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )^{3} + \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) + \sin \left (\frac {4}{9} \, \pi \right )\right )} \arctan \left (\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {4}{9} \, \pi \right ) + 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {4}{9} \, \pi \right )}\right ) + \frac {1}{9} \, {\left (2 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{4} - 12 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{2} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 2 \, \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right )^{4} + 8 \, \cos \left (\frac {2}{9} \, \pi \right )^{3} \sin \left (\frac {2}{9} \, \pi \right ) - 8 \, \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )^{3} + \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) + \sin \left (\frac {2}{9} \, \pi \right )\right )} \arctan \left (\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {2}{9} \, \pi \right ) + 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {2}{9} \, \pi \right )}\right ) + \frac {1}{9} \, {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{4} - 12 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} \sin \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right )^{4} - 8 \, \cos \left (\frac {1}{9} \, \pi \right )^{3} \sin \left (\frac {1}{9} \, \pi \right ) + 8 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )^{3} - \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) + \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (-\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {1}{9} \, \pi \right ) - 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {1}{9} \, \pi \right )}\right ) + \frac {1}{18} \, {\left (8 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{3} \sin \left (\frac {4}{9} \, \pi \right ) - 8 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )^{3} - 2 \, \cos \left (\frac {4}{9} \, \pi \right )^{4} + 12 \, \cos \left (\frac {4}{9} \, \pi \right )^{2} \sin \left (\frac {4}{9} \, \pi \right )^{2} - 2 \, \sin \left (\frac {4}{9} \, \pi \right )^{4} + \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right ) - \cos \left (\frac {4}{9} \, \pi \right )\right )} \log \left ({\left (-i \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) - \cos \left (\frac {4}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) + \frac {1}{18} \, {\left (8 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{3} \sin \left (\frac {2}{9} \, \pi \right ) - 8 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )^{3} - 2 \, \cos \left (\frac {2}{9} \, \pi \right )^{4} + 12 \, \cos \left (\frac {2}{9} \, \pi \right )^{2} \sin \left (\frac {2}{9} \, \pi \right )^{2} - 2 \, \sin \left (\frac {2}{9} \, \pi \right )^{4} + \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right ) - \cos \left (\frac {2}{9} \, \pi \right )\right )} \log \left ({\left (-i \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) - \cos \left (\frac {2}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) - \frac {1}{18} \, {\left (8 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{3} \sin \left (\frac {1}{9} \, \pi \right ) - 8 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )^{3} + 2 \, \cos \left (\frac {1}{9} \, \pi \right )^{4} - 12 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} \sin \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \sin \left (\frac {1}{9} \, \pi \right )^{4} - \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right ) - \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left ({\left (i \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) + \cos \left (\frac {1}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) - \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.40, size = 332, normalized size = 0.79 \begin {gather*} \frac {\ln \left (x+\frac {2^{2/3}\,3^{5/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\frac {2^{2/3}\,3^{5/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {1}{2\,x^2}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,3^{1/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{2}+\frac {2^{2/3}\,3^{1/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{4/3}}{12}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,3^{1/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{2}+\frac {2^{2/3}\,3^{1/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{4/3}}{12}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,3^{1/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{4}-\frac {2^{2/3}\,3^{5/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{12}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,3^{1/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{4}+\frac {2^{2/3}\,3^{5/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{12}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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