Optimal. Leaf size=237 \[ -\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right ),-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right ),-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}}+\frac{1}{5} \sqrt{x^8+1} x \]
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Rubi [A] time = 0.0614129, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {321, 226, 1883} \[ \frac{1}{5} \sqrt{x^8+1} x-\frac{\sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{\sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 226
Rule 1883
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{1+x^8}} \, dx &=\frac{1}{5} x \sqrt{1+x^8}-\frac{1}{5} \int \frac{1}{\sqrt{1+x^8}} \, dx\\ &=\frac{1}{5} x \sqrt{1+x^8}-\frac{1}{10} \int \frac{1-x^2}{\sqrt{1+x^8}} \, dx-\frac{1}{10} \int \frac{1+x^2}{\sqrt{1+x^8}} \, dx\\ &=\frac{1}{5} x \sqrt{1+x^8}-\frac{x^3 \sqrt{\frac{\left (1+x^2\right )^2}{x^2}} \sqrt{-\frac{1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2}-2 x^2+\sqrt{2} x^4}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1+x^2\right ) \sqrt{1+x^8}}+\frac{x^3 \sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2}+2 x^2+\sqrt{2} x^4}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{10 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{1+x^8}}\\ \end{align*}
Mathematica [C] time = 0.0064052, size = 32, normalized size = 0.14 \[ \frac{1}{5} x \left (\sqrt{x^8+1}-\, _2F_1\left (\frac{1}{8},\frac{1}{2};\frac{9}{8};-x^8\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.03, size = 26, normalized size = 0.1 \begin{align*}{\frac{x}{5}\sqrt{{x}^{8}+1}}-{\frac{x}{5}{\mbox{$_2$F$_1$}({\frac{1}{8}},{\frac{1}{2}};\,{\frac{9}{8}};\,-{x}^{8})}} \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{x^{8}}{\sqrt{x^{8} + 1}}\,{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{x^{8}}{\sqrt{x^{8} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.79127, size = 29, normalized size = 0.12 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{8} \\ \frac{17}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{17}{8}\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{x^{8}}{\sqrt{x^{8} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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