Optimal. Leaf size=138 \[ -\frac{\log \left (x^2-\sqrt [6]{2} \sqrt{3} x+\sqrt [3]{2}\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\log \left (x^2+\sqrt [6]{2} \sqrt{3} x+\sqrt [3]{2}\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac{\tan ^{-1}\left (\sqrt{3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac{\tan ^{-1}\left (2^{5/6} x+\sqrt{3}\right )}{6\ 2^{5/6}} \]
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Rubi [A] time = 0.281148, 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 = {209, 634, 618, 204, 628, 203} \[ -\frac{\log \left (x^2-\sqrt [6]{2} \sqrt{3} x+\sqrt [3]{2}\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\log \left (x^2+\sqrt [6]{2} \sqrt{3} x+\sqrt [3]{2}\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac{\tan ^{-1}\left (\sqrt{3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac{\tan ^{-1}\left (2^{5/6} x+\sqrt{3}\right )}{6\ 2^{5/6}} \]
Antiderivative was successfully verified.
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Rule 209
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{2+x^6} \, dx &=\frac{\int \frac{\sqrt [6]{2}-\frac{\sqrt{3} x}{2}}{\sqrt [3]{2}-\sqrt [6]{2} \sqrt{3} x+x^2} \, dx}{3\ 2^{5/6}}+\frac{\int \frac{\sqrt [6]{2}+\frac{\sqrt{3} x}{2}}{\sqrt [3]{2}+\sqrt [6]{2} \sqrt{3} x+x^2} \, dx}{3\ 2^{5/6}}+\frac{\int \frac{1}{\sqrt [3]{2}+x^2} \, dx}{3\ 2^{2/3}}\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac{\int \frac{1}{\sqrt [3]{2}-\sqrt [6]{2} \sqrt{3} x+x^2} \, dx}{12\ 2^{2/3}}+\frac{\int \frac{1}{\sqrt [3]{2}+\sqrt [6]{2} \sqrt{3} x+x^2} \, dx}{12\ 2^{2/3}}-\frac{\int \frac{-\sqrt [6]{2} \sqrt{3}+2 x}{\sqrt [3]{2}-\sqrt [6]{2} \sqrt{3} x+x^2} \, dx}{4\ 2^{5/6} \sqrt{3}}+\frac{\int \frac{\sqrt [6]{2} \sqrt{3}+2 x}{\sqrt [3]{2}+\sqrt [6]{2} \sqrt{3} x+x^2} \, dx}{4\ 2^{5/6} \sqrt{3}}\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt{3} x+x^2\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt{3} x+x^2\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2^{5/6} x}{\sqrt{3}}\right )}{6\ 2^{5/6} \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2^{5/6} x}{\sqrt{3}}\right )}{6\ 2^{5/6} \sqrt{3}}\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac{\tan ^{-1}\left (\sqrt{3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac{\tan ^{-1}\left (\sqrt{3}+2^{5/6} x\right )}{6\ 2^{5/6}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt{3} x+x^2\right )}{4\ 2^{5/6} \sqrt{3}}+\frac{\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt{3} x+x^2\right )}{4\ 2^{5/6} \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0270939, size = 115, normalized size = 0.83 \[ \frac{-\sqrt{3} \log \left (2^{2/3} x^2-2^{5/6} \sqrt{3} x+2\right )+\sqrt{3} \log \left (2^{2/3} x^2+2^{5/6} \sqrt{3} x+2\right )+4 \tan ^{-1}\left (\frac{x}{\sqrt [6]{2}}\right )-2 \tan ^{-1}\left (\sqrt{3}-2^{5/6} x\right )+2 \tan ^{-1}\left (2^{5/6} x+\sqrt{3}\right )}{12\ 2^{5/6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 95, normalized size = 0.7 \begin{align*}{\frac{\sqrt [6]{2}}{6}\arctan \left ({\frac{x{2}^{{\frac{5}{6}}}}{2}} \right ) }+{\frac{\arctan \left ( x{2}^{{\frac{5}{6}}}-\sqrt{3} \right ) \sqrt [6]{2}}{12}}+{\frac{\arctan \left ( x{2}^{{\frac{5}{6}}}+\sqrt{3} \right ) \sqrt [6]{2}}{12}}-{\frac{\ln \left ( \sqrt [3]{2}+{x}^{2}-\sqrt [6]{2}x\sqrt{3} \right ) \sqrt [6]{2}\sqrt{3}}{24}}+{\frac{\ln \left ( \sqrt [3]{2}+{x}^{2}+\sqrt [6]{2}x\sqrt{3} \right ) \sqrt [6]{2}\sqrt{3}}{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43452, size = 144, normalized size = 1.04 \begin{align*} \frac{1}{24} \, \sqrt{3} 2^{\frac{1}{6}} \log \left (x^{2} + \sqrt{3} 2^{\frac{1}{6}} x + 2^{\frac{1}{3}}\right ) - \frac{1}{24} \, \sqrt{3} 2^{\frac{1}{6}} \log \left (x^{2} - \sqrt{3} 2^{\frac{1}{6}} x + 2^{\frac{1}{3}}\right ) + \frac{1}{12} \cdot 2^{\frac{1}{6}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{5}{6}}{\left (2 \, x + \sqrt{3} 2^{\frac{1}{6}}\right )}\right ) + \frac{1}{12} \cdot 2^{\frac{1}{6}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{5}{6}}{\left (2 \, x - \sqrt{3} 2^{\frac{1}{6}}\right )}\right ) + \frac{1}{6} \cdot 2^{\frac{1}{6}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{5}{6}} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13277, size = 593, normalized size = 4.3 \begin{align*} \frac{1}{384} \cdot 32^{\frac{5}{6}} \sqrt{3} \log \left (32^{\frac{5}{6}} \sqrt{3} x + 16 \, x^{2} + 8 \cdot 4^{\frac{2}{3}}\right ) - \frac{1}{384} \cdot 32^{\frac{5}{6}} \sqrt{3} \log \left (-32^{\frac{5}{6}} \sqrt{3} x + 16 \, x^{2} + 8 \cdot 4^{\frac{2}{3}}\right ) - \frac{1}{48} \cdot 32^{\frac{5}{6}} \arctan \left (\frac{1}{4} \cdot 32^{\frac{1}{6}} \sqrt{2} \sqrt{2 \, x^{2} + 4^{\frac{2}{3}}} - \frac{1}{2} \cdot 32^{\frac{1}{6}} x\right ) - \frac{1}{96} \cdot 32^{\frac{5}{6}} \arctan \left (-32^{\frac{1}{6}} x + \frac{1}{4} \cdot 32^{\frac{1}{6}} \sqrt{32^{\frac{5}{6}} \sqrt{3} x + 16 \, x^{2} + 8 \cdot 4^{\frac{2}{3}}} - \sqrt{3}\right ) - \frac{1}{96} \cdot 32^{\frac{5}{6}} \arctan \left (-32^{\frac{1}{6}} x + \frac{1}{4} \cdot 32^{\frac{1}{6}} \sqrt{-32^{\frac{5}{6}} \sqrt{3} x + 16 \, x^{2} + 8 \cdot 4^{\frac{2}{3}}} + \sqrt{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.255566, 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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