Optimal. Leaf size=35 \[ \frac{1}{8} x^4 \sqrt{x^8-2}-\frac{1}{4} \tanh ^{-1}\left (\frac{x^4}{\sqrt{x^8-2}}\right ) \]
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Rubi [A] time = 0.0120813, 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} \[ \frac{1}{8} x^4 \sqrt{x^8-2}-\frac{1}{4} \tanh ^{-1}\left (\frac{x^4}{\sqrt{x^8-2}}\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 \sqrt{-2+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \sqrt{-2+x^2} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{-2+x^8}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{-2+x^8}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x^4}{\sqrt{-2+x^8}}\right )\\ &=\frac{1}{8} x^4 \sqrt{-2+x^8}-\frac{1}{4} \tanh ^{-1}\left (\frac{x^4}{\sqrt{-2+x^8}}\right )\\ \end{align*}
Mathematica [A] time = 0.017889, size = 50, normalized size = 1.43 \[ \frac{\left (x^8-2\right ) \left (\sqrt{2-x^8} x^4+2 \sin ^{-1}\left (\frac{x^4}{\sqrt{2}}\right )\right )}{8 \sqrt{-\left (x^8-2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.046, size = 47, normalized size = 1.3 \begin{align*}{\frac{{x}^{4}}{8}\sqrt{{x}^{8}-2}}-{\frac{1}{4}\sqrt{-{\it signum} \left ( -1+{\frac{{x}^{8}}{2}} \right ) }\arcsin \left ({\frac{{x}^{4}\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{\it signum} \left ( -1+{\frac{{x}^{8}}{2}} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.966931, size = 78, normalized size = 2.23 \begin{align*} -\frac{\sqrt{x^{8} - 2}}{4 \, x^{4}{\left (\frac{x^{8} - 2}{x^{8}} - 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} - 2}}{x^{4}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x^{8} - 2}}{x^{4}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28885, size = 74, normalized size = 2.11 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} - 2} x^{4} + \frac{1}{4} \, \log \left (-x^{4} + \sqrt{x^{8} - 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.5826, size = 90, normalized size = 2.57 \begin{align*} \begin{cases} \frac{x^{12}}{8 \sqrt{x^{8} - 2}} - \frac{x^{4}}{4 \sqrt{x^{8} - 2}} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} x^{4}}{2} \right )}}{4} & \text{for}\: \frac{\left |{x^{8}}\right |}{2} > 1 \\- \frac{i x^{12}}{8 \sqrt{2 - x^{8}}} + \frac{i x^{4}}{4 \sqrt{2 - x^{8}}} + \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} x^{4}}{2} \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16387, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} - 2} x^{4} + \frac{1}{4} \, \log \left (x^{4} - \sqrt{x^{8} - 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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