Optimal. Leaf size=35 \[ \frac{1}{6} x^3 \sqrt{x^6-2}-\frac{1}{3} \tanh ^{-1}\left (\frac{x^3}{\sqrt{x^6-2}}\right ) \]
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Rubi [A] time = 0.0118392, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 195, 217, 206} \[ \frac{1}{6} x^3 \sqrt{x^6-2}-\frac{1}{3} \tanh ^{-1}\left (\frac{x^3}{\sqrt{x^6-2}}\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \sqrt{-2+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \sqrt{-2+x^2} \, dx,x,x^3\right )\\ &=\frac{1}{6} x^3 \sqrt{-2+x^6}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+x^2}} \, dx,x,x^3\right )\\ &=\frac{1}{6} x^3 \sqrt{-2+x^6}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x^3}{\sqrt{-2+x^6}}\right )\\ &=\frac{1}{6} x^3 \sqrt{-2+x^6}-\frac{1}{3} \tanh ^{-1}\left (\frac{x^3}{\sqrt{-2+x^6}}\right )\\ \end{align*}
Mathematica [A] time = 0.0175528, size = 50, normalized size = 1.43 \[ \frac{\left (x^6-2\right ) \left (\sqrt{2-x^6} x^3+2 \sin ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )\right )}{6 \sqrt{-\left (x^6-2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.043, size = 47, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}}{6}\sqrt{{x}^{6}-2}}-{\frac{1}{3}\sqrt{-{\it signum} \left ( -1+{\frac{{x}^{6}}{2}} \right ) }\arcsin \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{\it signum} \left ( -1+{\frac{{x}^{6}}{2}} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02271, size = 78, normalized size = 2.23 \begin{align*} -\frac{\sqrt{x^{6} - 2}}{3 \, x^{3}{\left (\frac{x^{6} - 2}{x^{6}} - 1\right )}} - \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} - 2}}{x^{3}} + 1\right ) + \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} - 2}}{x^{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46163, size = 74, normalized size = 2.11 \begin{align*} \frac{1}{6} \, \sqrt{x^{6} - 2} x^{3} + \frac{1}{3} \, \log \left (-x^{3} + \sqrt{x^{6} - 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.54141, size = 90, normalized size = 2.57 \begin{align*} \begin{cases} \frac{x^{9}}{6 \sqrt{x^{6} - 2}} - \frac{x^{3}}{3 \sqrt{x^{6} - 2}} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} & \text{for}\: \frac{\left |{x^{6}}\right |}{2} > 1 \\- \frac{i x^{9}}{6 \sqrt{2 - x^{6}}} + \frac{i x^{3}}{3 \sqrt{2 - x^{6}}} + \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14012, size = 41, normalized size = 1.17 \begin{align*} \frac{1}{6} \, \sqrt{x^{6} - 2} x^{3} + \frac{1}{3} \, \log \left ({\left | -x^{3} + \sqrt{x^{6} - 2} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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