Optimal. Leaf size=76 \[ -\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{12 \sqrt{x^4+1}}-\frac{5 \sqrt{x^4+1}}{6 x^3}+\frac{1}{2 x^3 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0146343, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 220} \[ -\frac{5 \sqrt{x^4+1}}{6 x^3}+\frac{1}{2 x^3 \sqrt{x^4+1}}-\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{12 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^3 \sqrt{1+x^4}}+\frac{5}{2} \int \frac{1}{x^4 \sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^3 \sqrt{1+x^4}}-\frac{5 \sqrt{1+x^4}}{6 x^3}-\frac{5}{6} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x^3 \sqrt{1+x^4}}-\frac{5 \sqrt{1+x^4}}{6 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 )}{12 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0025907, size = 22, normalized size = 0.29 \[ -\frac{\, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{1}{4};-x^4\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 84, normalized size = 1.1 \begin{align*} -{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{4}+1}}-{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{3\,\sqrt{2}+3\,i\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^{4}}\,{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} + 2 \, x^{8} + x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.964751, size = 32, normalized size = 0.42 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{3} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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