Optimal. Leaf size=180 \[ -\frac {\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \]
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Rubi [A] time = 0.28, 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 = {210, 634, 618, 204, 628, 206} \[ -\frac {\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac {\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {2^{5/6} x}{\sqrt [3]{3}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [6]{\frac {3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 210
Rule 618
Rule 628
Rule 634
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 {\operatorname {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 {\operatorname {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 = 0.08, size = 162, normalized size = 0.90 \[ \frac {\sqrt {3} \left (-\log \left (2^{2/3} \sqrt [3]{3} x^2-2^{5/6} \sqrt [6]{3} x+2\right )+\log \left (2^{2/3} \sqrt [3]{3} x^2+2^{5/6} \sqrt [6]{3} x+2\right )-2 \log \left (2-2^{5/6} \sqrt [6]{3} x\right )+2 \log \left (2^{5/6} \sqrt [6]{3} x+2\right )\right )+6 \tan ^{-1}\left (\frac {2^{5/6} x}{\sqrt [3]{3}}+\frac {1}{\sqrt {3}}\right )+6 \tan ^{-1}\left (\frac {2^{5/6} \sqrt [6]{3} x-1}{\sqrt {3}}\right )}{12\ 2^{5/6} 3^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 169, normalized size = 0.94 \[ -\frac {1}{288} \cdot 96^{\frac {5}{6}} \sqrt {3} \arctan \left (-\frac {1}{3} \cdot 96^{\frac {1}{6}} \sqrt {3} x + \frac {1}{12} \cdot 96^{\frac {1}{6}} \sqrt {48 \, x^{2} + 96^{\frac {5}{6}} x + 8 \cdot 12^{\frac {2}{3}}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{288} \cdot 96^{\frac {5}{6}} \sqrt {3} \arctan \left (-\frac {1}{3} \cdot 96^{\frac {1}{6}} \sqrt {3} x + \frac {1}{12} \cdot 96^{\frac {1}{6}} \sqrt {48 \, x^{2} - 96^{\frac {5}{6}} x + 8 \cdot 12^{\frac {2}{3}}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{1152} \cdot 96^{\frac {5}{6}} \log \left (48 \, x^{2} + 96^{\frac {5}{6}} x + 8 \cdot 12^{\frac {2}{3}}\right ) - \frac {1}{1152} \cdot 96^{\frac {5}{6}} \log \left (48 \, x^{2} - 96^{\frac {5}{6}} x + 8 \cdot 12^{\frac {2}{3}}\right ) + \frac {1}{576} \cdot 96^{\frac {5}{6}} \log \left (48 \, x + 96^{\frac {5}{6}}\right ) - \frac {1}{576} \cdot 96^{\frac {5}{6}} \log \left (48 \, x - 96^{\frac {5}{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 114, normalized size = 0.63 \[ \frac {1}{12} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{2} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {5}{6}} {\left (2 \, x + \left (\frac {2}{3}\right )^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{2} \, \sqrt {3} \left (\frac {2}{3}\right )^{\frac {5}{6}} {\left (2 \, x - \left (\frac {2}{3}\right )^{\frac {1}{6}}\right )}\right ) + \frac {1}{72} \cdot 486^{\frac {1}{6}} \log \left (x^{2} + \left (\frac {2}{3}\right )^{\frac {1}{6}} x + \left (\frac {2}{3}\right )^{\frac {1}{3}}\right ) - \frac {1}{72} \cdot 486^{\frac {1}{6}} \log \left (x^{2} - \left (\frac {2}{3}\right )^{\frac {1}{6}} x + \left (\frac {2}{3}\right )^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 486^{\frac {1}{6}} \log \left ({\left | x + \left (\frac {2}{3}\right )^{\frac {1}{6}} \right |}\right ) - \frac {1}{36} \cdot 486^{\frac {1}{6}} \log \left ({\left | x - \left (\frac {2}{3}\right )^{\frac {1}{6}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 228, normalized size = 1.27 \[ -\frac {2^{\frac {1}{6}} 3^{\frac {1}{3}} \arctan \left (\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}-\frac {\sqrt {2}\, \sqrt {6}}{6}\right )}{36}+\frac {2^{\frac {5}{6}} 3^{\frac {2}{3}} 12^{\frac {2}{3}} \arctan \left (\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}-\frac {\sqrt {2}\, \sqrt {6}}{6}\right )}{108}-\frac {2^{\frac {1}{6}} 3^{\frac {1}{3}} \arctan \left (\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}+\frac {\sqrt {2}\, \sqrt {6}}{6}\right )}{36}+\frac {2^{\frac {5}{6}} 3^{\frac {2}{3}} 12^{\frac {2}{3}} \arctan \left (\frac {\sqrt {2}\, 12^{\frac {2}{3}} x}{6}+\frac {\sqrt {2}\, \sqrt {6}}{6}\right )}{108}-\frac {\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (6 x -\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}}\right )}{72}+\frac {\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (6 x +\sqrt {6}\, 3^{\frac {1}{3}} 2^{\frac {2}{3}}\right )}{72}-\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \sqrt {6}\, \ln \left (6 x^{2}-\sqrt {6}\, 12^{\frac {1}{3}} x +12^{\frac {2}{3}}\right )}{144}+\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \sqrt {6}\, \ln \left (6 x^{2}+\sqrt {6}\, 12^{\frac {1}{3}} x +12^{\frac {2}{3}}\right )}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 224, normalized size = 1.24 \[ \frac {1}{12} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 3^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {1}{3}\right )^{\frac {2}{3}} {\left (2 \, x + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {2}{3}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 3^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {1}{3}\right )^{\frac {2}{3}} {\left (2 \, x - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x^{2} + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}} x + \left (\frac {1}{3}\right )^{\frac {2}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {2}{3}}\right ) - \frac {1}{24} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x^{2} - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}} x + \left (\frac {1}{3}\right )^{\frac {2}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{6}} 2^{\frac {1}{6}} \left (\frac {1}{3}\right )^{\frac {1}{3}} \log \left (x - \left (\frac {1}{3}\right )^{\frac {1}{3}} \left (\sqrt {3} \sqrt {2}\right )^{\frac {1}{3}}\right ) \]
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 \[ \frac {{486}^{1/6}\,\mathrm {atanh}\left (\frac {{486}^{5/6}\,x}{162}\right )}{18}-\frac {2^{1/6}\,\mathrm {atanh}\left (\frac {2^{1/6}\,3^{1/3}\,x\,1{}\mathrm {i}}{162\,\left (\frac {2^{1/3}\,3^{2/3}}{486}-\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}+\frac {2^{1/6}\,3^{5/6}\,x}{486\,\left (\frac {2^{1/3}\,3^{2/3}}{486}-\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}\right )\,\left (3^{5/6}+3^{1/3}\,3{}\mathrm {i}\right )}{36}-\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,3^{1/3}\,x}{162\,\left (\frac {2^{1/3}\,3^{2/3}}{486}+\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}+\frac {2^{1/6}\,3^{5/6}\,x\,1{}\mathrm {i}}{486\,\left (\frac {2^{1/3}\,3^{2/3}}{486}+\frac {2^{1/3}\,3^{1/6}\,1{}\mathrm {i}}{162}\right )}\right )\,\left (-3^{5/6}+3^{1/3}\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 15, normalized size = 0.08 \[ - \operatorname {RootSum} {\left (4478976 t^{6} - 1, \left (t \mapsto t \log {\left (- 12 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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