Optimal. Leaf size=239 \[ -\frac{3 \sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right ),-2 \left (1-\sqrt{2}\right )\right )}{14 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{3 \sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right ),-2 \left (1-\sqrt{2}\right )\right )}{14 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}}-\frac{\sqrt{x^8+1}}{7 x^7} \]
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Rubi [A] time = 0.0611552, antiderivative size = 239, 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 = {325, 226, 1883} \[ -\frac{\sqrt{x^8+1}}{7 x^7}-\frac{3 \sqrt{\frac{\left (x^2+1\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{-\frac{\sqrt{2} x^4-2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{14 \sqrt{2+\sqrt{2}} \left (x^2+1\right ) \sqrt{x^8+1}}+\frac{3 \sqrt{-\frac{\left (1-x^2\right )^2}{x^2}} \sqrt{-\frac{x^8+1}{x^4}} x^3 F\left (\sin ^{-1}\left (\frac{1}{2} \sqrt{\frac{\sqrt{2} x^4+2 x^2+\sqrt{2}}{x^2}}\right )|-2 \left (1-\sqrt{2}\right )\right )}{14 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 226
Rule 1883
Rubi steps
\begin{align*} \int \frac{1}{x^8 \sqrt{1+x^8}} \, dx &=-\frac{\sqrt{1+x^8}}{7 x^7}-\frac{3}{7} \int \frac{1}{\sqrt{1+x^8}} \, dx\\ &=-\frac{\sqrt{1+x^8}}{7 x^7}-\frac{3}{14} \int \frac{1-x^2}{\sqrt{1+x^8}} \, dx-\frac{3}{14} \int \frac{1+x^2}{\sqrt{1+x^8}} \, dx\\ &=-\frac{\sqrt{1+x^8}}{7 x^7}-\frac{3 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 )}{14 \sqrt{2+\sqrt{2}} \left (1+x^2\right ) \sqrt{1+x^8}}+\frac{3 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 )}{14 \sqrt{2+\sqrt{2}} \left (1-x^2\right ) \sqrt{1+x^8}}\\ \end{align*}
Mathematica [C] time = 0.0024397, size = 22, normalized size = 0.09 \[ -\frac{\, _2F_1\left (-\frac{7}{8},\frac{1}{2};\frac{1}{8};-x^8\right )}{7 x^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 28, normalized size = 0.1 \begin{align*} -{\frac{1}{7\,{x}^{7}}\sqrt{{x}^{8}+1}}-{\frac{3\,x}{7}{\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{1}{\sqrt{x^{8} + 1} x^{8}}\,{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^{8} + 1}}{x^{16} + x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.993806, size = 32, normalized size = 0.13 \begin{align*} \frac{\Gamma \left (- \frac{7}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{8}, \frac{1}{2} \\ \frac{1}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x^{7} \Gamma \left (\frac{1}{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{1}{\sqrt{x^{8} + 1} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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