Optimal. Leaf size=180 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{6}+2 \sqrt [3]{3} x}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}} \]
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Rubi [A]
time = 0.20, antiderivative size = 167, normalized size of antiderivative = 0.93, number of steps
used = 10, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {216, 648, 632,
210, 642, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\text {ArcTan}\left (\frac {2^{5/6} x}{\sqrt [3]{3}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{5/6} 3^{2/3}}-\frac {\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{2-3 x^6} \, dx &=\frac {\int \frac {\sqrt [6]{2}-\frac {\sqrt [6]{3} x}{2}}{\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{3\ 2^{5/6}}+\frac {\int \frac {\sqrt [6]{2}+\frac {\sqrt [6]{3} x}{2}}{\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{3\ 2^{5/6}}+\frac {\int \frac {1}{\sqrt [3]{2}-\sqrt [3]{3} x^2} \, dx}{3\ 2^{2/3}}\\ &=\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}+\frac {\int \frac {1}{\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{4\ 2^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{4\ 2^{2/3}}-\frac {\int \frac {-\sqrt [6]{6}+2 \sqrt [3]{3} x}{\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\int \frac {\sqrt [6]{6}+2 \sqrt [3]{3} x}{\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2} \, dx}{12\ 2^{5/6} \sqrt [6]{3}}\\ &=\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{5/6} \sqrt [6]{3} x\right )}{2\ 2^{5/6} \sqrt [6]{3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{5/6} \sqrt [6]{3} x\right )}{2\ 2^{5/6} \sqrt [6]{3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{6} x+\sqrt [3]{3} x^2\right )}{12\ 2^{5/6} \sqrt [6]{3}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 162, normalized size = 0.90 \begin {gather*} \frac {6 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} x}{\sqrt [3]{3}}\right )+6 \tan ^{-1}\left (\frac {-1+2^{5/6} \sqrt [6]{3} x}{\sqrt {3}}\right )+\sqrt {3} \left (-2 \log \left (2-2^{5/6} \sqrt [6]{3} x\right )+2 \log \left (2+2^{5/6} \sqrt [6]{3} x\right )-\log \left (2-2^{5/6} \sqrt [6]{3} x+2^{2/3} \sqrt [3]{3} x^2\right )+\log \left (2+2^{5/6} \sqrt [6]{3} x+2^{2/3} \sqrt [3]{3} x^2\right )\right )}{12\ 2^{5/6} 3^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 228, normalized size = 1.27
method | result | size |
risch | \(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (3 \textit {\_Z}^{6}-2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{18}\) | \(24\) |
meijerg | \(-\frac {96^{\frac {5}{6}} x \left (\ln \left (1-\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )-\ln \left (1+\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )+\frac {\ln \left (1-\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {3^{\frac {2}{3}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4-3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {3^{\frac {2}{3}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4+3^{\frac {1}{6}} 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{576 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(181\) |
default | \(-\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \sqrt {6}\, \ln \left (-x \sqrt {6}\, 12^{\frac {1}{3}}+12^{\frac {2}{3}}+6 x^{2}\right )}{144}-\frac {2^{\frac {1}{6}} 3^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{36}+\frac {2^{\frac {5}{6}} 3^{\frac {2}{3}} 12^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{108}+\frac {\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}}+6 x \right )}{72}+\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \sqrt {6}\, \ln \left (x \sqrt {6}\, 12^{\frac {1}{3}}+12^{\frac {2}{3}}+6 x^{2}\right )}{144}-\frac {2^{\frac {1}{6}} 3^{\frac {1}{3}} \arctan \left (\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{36}+\frac {2^{\frac {5}{6}} 3^{\frac {2}{3}} 12^{\frac {2}{3}} \arctan \left (\frac {\sqrt {2}\, \sqrt {6}}{6}+\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}\right )}{108}-\frac {\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (-\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}}+6 x \right )}{72}\) | \(228\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 224, normalized size = 1.24 \begin {gather*} \frac {1}{12} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 3^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {1}{3}\right )^{\frac {2}{3}} {\left (2 \, x + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 3^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {1}{3}\right )^{\frac {2}{3}} {\left (2 \, x - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x^{2} + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}} x + \left (\frac {1}{3}\right )^{\frac {2}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {2}{3}}\right ) - \frac {1}{24} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x^{2} - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}} x + \left (\frac {1}{3}\right )^{\frac {2}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 170, normalized size = 0.94 \begin {gather*} -\frac {1}{288} \cdot 96^{\frac {5}{6}} \sqrt {3} \arctan \left (-\frac {1}{3} \cdot 96^{\frac {1}{6}} \sqrt {3} x + \frac {1}{12} \cdot 96^{\frac {1}{6}} \sqrt {48 \, x^{2} + 96^{\frac {5}{6}} x + 8 \cdot 12^{\frac {2}{3}}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{288} \cdot 96^{\frac {5}{6}} \sqrt {3} \arctan \left (-\frac {1}{3} \cdot 96^{\frac {1}{6}} \sqrt {3} x + \frac {1}{12} \cdot 96^{\frac {1}{6}} \sqrt {48 \, x^{2} - 96^{\frac {5}{6}} x + 8 \cdot 12^{\frac {2}{3}}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{1152} \cdot 96^{\frac {5}{6}} \log \left (9216 \, x^{2} + 192 \cdot 96^{\frac {5}{6}} x + 1536 \cdot 12^{\frac {2}{3}}\right ) - \frac {1}{1152} \cdot 96^{\frac {5}{6}} \log \left (9216 \, x^{2} - 192 \cdot 96^{\frac {5}{6}} x + 1536 \cdot 12^{\frac {2}{3}}\right ) + \frac {1}{576} \cdot 96^{\frac {5}{6}} \log \left (48 \, x + 96^{\frac {5}{6}}\right ) - \frac {1}{576} \cdot 96^{\frac {5}{6}} \log \left (48 \, x - 96^{\frac {5}{6}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 15, normalized size = 0.08 \begin {gather*} - \operatorname {RootSum} {\left (4478976 t^{6} - 1, \left ( t \mapsto t \log {\left (- 12 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.97, size = 114, normalized size = 0.63 \begin {gather*} \frac {1}{12} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{2} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {5}{6}} {\left (2 \, x + \left (\frac {2}{3}\right )^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{2} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {5}{6}} {\left (2 \, x - \left (\frac {2}{3}\right )^{\frac {1}{6}}\right )}\right ) + \frac {1}{72} \cdot 486^{\frac {1}{6}} \log \left (x^{2} + \left (\frac {2}{3}\right )^{\frac {1}{6}} x + \left (\frac {2}{3}\right )^{\frac {1}{3}}\right ) - \frac {1}{72} \cdot 486^{\frac {1}{6}} \log \left (x^{2} - \left (\frac {2}{3}\right )^{\frac {1}{6}} x + \left (\frac {2}{3}\right )^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 486^{\frac {1}{6}} \log \left ({\left | x + \left (\frac {2}{3}\right )^{\frac {1}{6}} \right |}\right ) - \frac {1}{36} \cdot 486^{\frac {1}{6}} \log \left ({\left | x - \left (\frac {2}{3}\right )^{\frac {1}{6}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 164, normalized size = 0.91 \begin {gather*} \frac {{486}^{1/6}\,\mathrm {atanh}\left (\frac {{486}^{5/6}\,x}{162}\right )}{18}-\frac {2^{1/6}\,\mathrm {atanh}\left (\frac {2^{1/6}\,3^{1/3}\,x\,1{}\mathrm {i}}{162\,\left (\frac {2^{1/3}\,3^{2/3}}{486}-\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}+\frac {2^{1/6}\,3^{5/6}\,x}{486\,\left (\frac {2^{1/3}\,3^{2/3}}{486}-\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}\right )\,\left (3^{5/6}+3^{1/3}\,3{}\mathrm {i}\right )}{36}-\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,3^{1/3}\,x}{162\,\left (\frac {2^{1/3}\,3^{2/3}}{486}+\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}+\frac {2^{1/6}\,3^{5/6}\,x\,1{}\mathrm {i}}{486\,\left (\frac {2^{1/3}\,3^{2/3}}{486}+\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}\right )\,\left (-3^{5/6}+3^{1/3}\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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