Optimal. Leaf size=138 \[ \frac{\log \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}-\frac{\log \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}+\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}} \]
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Rubi [A] time = 0.191552, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {210, 634, 618, 204, 628, 206} \[ \frac{\log \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}-\frac{\log \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}+\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}} \]
Antiderivative was successfully verified.
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Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{-2+x^6} \, dx &=-\frac{\int \frac{\sqrt [6]{2}-\frac{x}{2}}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx}{3\ 2^{5/6}}-\frac{\int \frac{\sqrt [6]{2}+\frac{x}{2}}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx}{3\ 2^{5/6}}-\frac{\int \frac{1}{\sqrt [3]{2}-x^2} \, dx}{3\ 2^{2/3}}\\ &=-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac{\int \frac{-\sqrt [6]{2}+2 x}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx}{12\ 2^{5/6}}-\frac{\int \frac{\sqrt [6]{2}+2 x}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx}{12\ 2^{5/6}}-\frac{\int \frac{1}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx}{4\ 2^{2/3}}-\frac{\int \frac{1}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx}{4\ 2^{2/3}}\\ &=-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2^{5/6} x\right )}{2\ 2^{5/6}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{5/6} x\right )}{2\ 2^{5/6}}\\ &=\frac{\tan ^{-1}\left (\frac{1-2^{5/6} x}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1+2^{5/6} x}{\sqrt{3}}\right )}{2\ 2^{5/6} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}\\ \end{align*}
Mathematica [A] time = 0.0401925, size = 122, normalized size = 0.88 \[ -\frac{-\log \left (2^{2/3} x^2-2^{5/6} x+2\right )+\log \left (2^{2/3} x^2+2^{5/6} x+2\right )-2 \log \left (2-2^{5/6} x\right )+2 \log \left (2^{5/6} x+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{5/6} x+1}{\sqrt{3}}\right )}{12\ 2^{5/6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 111, normalized size = 0.8 \begin{align*} -{\frac{\sqrt [6]{2}\ln \left ( x+\sqrt [6]{2} \right ) }{12}}+{\frac{\ln \left ( \sqrt [3]{2}-\sqrt [6]{2}x+{x}^{2} \right ) \sqrt [6]{2}}{24}}-{\frac{\sqrt{3}\sqrt [6]{2}}{12}\arctan \left ( -{\frac{\sqrt{3}}{3}}+{\frac{{2}^{{\frac{5}{6}}}x\sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( \sqrt [3]{2}+\sqrt [6]{2}x+{x}^{2} \right ) \sqrt [6]{2}}{24}}-{\frac{\sqrt{3}\sqrt [6]{2}}{12}\arctan \left ({\frac{\sqrt{3}}{3}}+{\frac{{2}^{{\frac{5}{6}}}x\sqrt{3}}{3}} \right ) }+{\frac{\sqrt [6]{2}\ln \left ( x-\sqrt [6]{2} \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4415, size = 151, normalized size = 1.09 \begin{align*} -\frac{1}{12} \, \sqrt{3} 2^{\frac{1}{6}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{5}{6}}{\left (2 \, x + 2^{\frac{1}{6}}\right )}\right ) - \frac{1}{12} \, \sqrt{3} 2^{\frac{1}{6}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{5}{6}}{\left (2 \, x - 2^{\frac{1}{6}}\right )}\right ) - \frac{1}{24} \cdot 2^{\frac{1}{6}} \log \left (x^{2} + 2^{\frac{1}{6}} x + 2^{\frac{1}{3}}\right ) + \frac{1}{24} \cdot 2^{\frac{1}{6}} \log \left (x^{2} - 2^{\frac{1}{6}} x + 2^{\frac{1}{3}}\right ) - \frac{1}{12} \cdot 2^{\frac{1}{6}} \log \left (x + 2^{\frac{1}{6}}\right ) + \frac{1}{12} \cdot 2^{\frac{1}{6}} \log \left (x - 2^{\frac{1}{6}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02239, size = 603, normalized size = 4.37 \begin{align*} \frac{1}{96} \cdot 32^{\frac{5}{6}} \sqrt{3} \arctan \left (-\frac{1}{3} \cdot 32^{\frac{1}{6}} \sqrt{3} x + \frac{1}{12} \cdot 32^{\frac{1}{6}} \sqrt{3} \sqrt{16 \, x^{2} + 32^{\frac{5}{6}} x + 8 \cdot 4^{\frac{2}{3}}} - \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{96} \cdot 32^{\frac{5}{6}} \sqrt{3} \arctan \left (-\frac{1}{3} \cdot 32^{\frac{1}{6}} \sqrt{3} x + \frac{1}{12} \cdot 32^{\frac{1}{6}} \sqrt{3} \sqrt{16 \, x^{2} - 32^{\frac{5}{6}} x + 8 \cdot 4^{\frac{2}{3}}} + \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{384} \cdot 32^{\frac{5}{6}} \log \left (16 \, x^{2} + 32^{\frac{5}{6}} x + 8 \cdot 4^{\frac{2}{3}}\right ) + \frac{1}{384} \cdot 32^{\frac{5}{6}} \log \left (16 \, x^{2} - 32^{\frac{5}{6}} x + 8 \cdot 4^{\frac{2}{3}}\right ) - \frac{1}{192} \cdot 32^{\frac{5}{6}} \log \left (16 \, x + 32^{\frac{5}{6}}\right ) + \frac{1}{192} \cdot 32^{\frac{5}{6}} \log \left (16 \, x - 32^{\frac{5}{6}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.571694, size = 14, normalized size = 0.1 \begin{align*} \operatorname{RootSum}{\left (1492992 t^{6} - 1, \left ( t \mapsto t \log{\left (- 12 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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