Optimal. Leaf size=156 \[ -\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{20 \sqrt{x^4+1}}-\frac{21 \sqrt{x^4+1} x}{10 \left (x^2+1\right )}+\frac{21 \sqrt{x^4+1}}{10 x}-\frac{7 \sqrt{x^4+1}}{10 x^5}+\frac{1}{2 \sqrt{x^4+1} x^5}+\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0344749, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {290, 325, 305, 220, 1196} \[ -\frac{21 \sqrt{x^4+1} x}{10 \left (x^2+1\right )}+\frac{21 \sqrt{x^4+1}}{10 x}-\frac{7 \sqrt{x^4+1}}{10 x^5}+\frac{1}{2 \sqrt{x^4+1} x^5}-\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{20 \sqrt{x^4+1}}+\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^5 \sqrt{1+x^4}}+\frac{7}{2} \int \frac{1}{x^6 \sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1+x^4}}-\frac{7 \sqrt{1+x^4}}{10 x^5}-\frac{21}{10} \int \frac{1}{x^2 \sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1+x^4}}-\frac{7 \sqrt{1+x^4}}{10 x^5}+\frac{21 \sqrt{1+x^4}}{10 x}-\frac{21}{10} \int \frac{x^2}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1+x^4}}-\frac{7 \sqrt{1+x^4}}{10 x^5}+\frac{21 \sqrt{1+x^4}}{10 x}-\frac{21}{10} \int \frac{1}{\sqrt{1+x^4}} \, dx+\frac{21}{10} \int \frac{1-x^2}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^5 \sqrt{1+x^4}}-\frac{7 \sqrt{1+x^4}}{10 x^5}+\frac{21 \sqrt{1+x^4}}{10 x}-\frac{21 x \sqrt{1+x^4}}{10 \left (1+x^2\right )}+\frac{21 \left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{1+x^4}}-\frac{21 \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 )}{20 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0028174, size = 22, normalized size = 0.14 \[ -\frac{\, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-x^4\right )}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 119, normalized size = 0.8 \begin{align*} -{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{4}+1}}+{\frac{8}{5\,x}\sqrt{{x}^{4}+1}}+{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{21\,i}{10}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{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}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{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^{14} + 2 \, x^{10} + x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.45795, size = 36, normalized size = 0.23 \begin{align*} \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ - \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{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}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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