Optimal. Leaf size=76 \[ \frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{42 \sqrt{x^4+1}}+\frac{5 \sqrt{x^4+1}}{21 x^3}-\frac{\sqrt{x^4+1}}{7 x^7} \]
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Rubi [A] time = 0.0155506, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 220} \[ \frac{5 \sqrt{x^4+1}}{21 x^3}-\frac{\sqrt{x^4+1}}{7 x^7}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^8 \sqrt{1+x^4}} \, dx &=-\frac{\sqrt{1+x^4}}{7 x^7}-\frac{5}{7} \int \frac{1}{x^4 \sqrt{1+x^4}} \, dx\\ &=-\frac{\sqrt{1+x^4}}{7 x^7}+\frac{5 \sqrt{1+x^4}}{21 x^3}+\frac{5}{21} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=-\frac{\sqrt{1+x^4}}{7 x^7}+\frac{5 \sqrt{1+x^4}}{21 x^3}+\frac{5 \left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0025629, size = 22, normalized size = 0.29 \[ -\frac{\, _2F_1\left (-\frac{7}{4},\frac{1}{2};-\frac{3}{4};-x^4\right )}{7 x^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 86, normalized size = 1.1 \begin{align*} -{\frac{1}{7\,{x}^{7}}\sqrt{{x}^{4}+1}}+{\frac{5}{21\,{x}^{3}}\sqrt{{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{{\frac{21\,\sqrt{2}}{2}}+{\frac{21\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \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^{4} + 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^{4} + 1}}{x^{12} + x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.41929, size = 36, normalized size = 0.47 \begin{align*} \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\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^{4} + 1} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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