Optimal. Leaf size=42 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]
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Rubi [A] time = 0.017391, antiderivative size = 42, 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 = {266, 51, 63, 207} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^7 \sqrt{2+x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{12 x^6}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{12 x^6}-\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+x^6}\right )\\ &=-\frac{\sqrt{2+x^6}}{12 x^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2+x^6}}{\sqrt{2}}\right )}{12 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0075629, size = 42, normalized size = 1. \[ \frac{\sqrt{2} x^6 \tanh ^{-1}\left (\sqrt{\frac{x^6}{2}+1}\right )-2 \sqrt{x^6+2}}{24 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 39, normalized size = 0.9 \begin{align*} -{\frac{1}{12\,{x}^{6}}\sqrt{{x}^{6}+2}}-{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50427, size = 63, normalized size = 1.5 \begin{align*} -\frac{1}{48} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{\sqrt{x^{6} + 2}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42565, size = 117, normalized size = 2.79 \begin{align*} \frac{\sqrt{2} x^{6} \log \left (\frac{x^{6} + 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) - 4 \, \sqrt{x^{6} + 2}}{48 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.98439, size = 31, normalized size = 0.74 \begin{align*} \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{24} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{12 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17579, size = 63, normalized size = 1.5 \begin{align*} -\frac{1}{48} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{\sqrt{x^{6} + 2}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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