Optimal. Leaf size=392 \[ -\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x \text{EllipticF}\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{2 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}-\frac{\sqrt{x^6+2}}{2 x}-\frac{\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
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Rubi [A] time = 0.0797777, antiderivative size = 392, 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 = {325, 308, 225, 1881} \[ \frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{2 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}-\frac{\sqrt{x^6+2}}{2 x}-\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{2+x^6}} \, dx &=-\frac{\sqrt{2+x^6}}{2 x}+\int \frac{x^4}{\sqrt{2+x^6}} \, dx\\ &=-\frac{\sqrt{2+x^6}}{2 x}-\frac{1}{2} \int \frac{2^{2/3} \left (-1+\sqrt{3}\right )-2 x^4}{\sqrt{2+x^6}} \, dx-\frac{\left (1-\sqrt{3}\right ) \int \frac{1}{\sqrt{2+x^6}} \, dx}{\sqrt [3]{2}}\\ &=-\frac{\sqrt{2+x^6}}{2 x}+\frac{\left (1+\sqrt{3}\right ) x \sqrt{2+x^6}}{2 \left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )}-\frac{\sqrt [4]{3} x \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}+\left (1-\sqrt{3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} \sqrt{\frac{x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} \sqrt{2+x^6}}-\frac{\left (1-\sqrt{3}\right ) x \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}+\left (1-\sqrt{3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} \sqrt{2+x^6}}\\ \end{align*}
Mathematica [C] time = 0.0042917, size = 27, normalized size = 0.07 \[ -\frac{\, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};-\frac{x^6}{2}\right )}{\sqrt{2} x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 33, normalized size = 0.1 \begin{align*} -{\frac{1}{2\,x}\sqrt{{x}^{6}+2}}+{\frac{{x}^{5}\sqrt{2}}{10}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{6}};\,-{\frac{{x}^{6}}{2}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} + 2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{6} + 2}}{x^{8} + 2 \, x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.603297, size = 37, normalized size = 0.09 \begin{align*} \frac{\sqrt{2} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{1}{2} \\ \frac{5}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 x \Gamma \left (\frac{5}{6}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} + 2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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