Optimal. Leaf size=140 \[ \frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{4 \sqrt{x^4+1}}+\frac{3 \sqrt{x^4+1} x}{2 \left (x^2+1\right )}-\frac{3 \sqrt{x^4+1}}{2 x}+\frac{1}{2 \sqrt{x^4+1} x}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0245369, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{3 \sqrt{x^4+1} x}{2 \left (x^2+1\right )}-\frac{3 \sqrt{x^4+1}}{2 x}+\frac{1}{2 \sqrt{x^4+1} x}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
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^2 \left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{2 x \sqrt{1+x^4}}+\frac{3}{2} \int \frac{1}{x^2 \sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x \sqrt{1+x^4}}-\frac{3 \sqrt{1+x^4}}{2 x}+\frac{3}{2} \int \frac{x^2}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x \sqrt{1+x^4}}-\frac{3 \sqrt{1+x^4}}{2 x}+\frac{3}{2} \int \frac{1}{\sqrt{1+x^4}} \, dx-\frac{3}{2} \int \frac{1-x^2}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{2 x \sqrt{1+x^4}}-\frac{3 \sqrt{1+x^4}}{2 x}+\frac{3 x \sqrt{1+x^4}}{2 \left (1+x^2\right )}-\frac{3 \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 )}{2 \sqrt{1+x^4}}+\frac{3 \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 )}{4 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0028402, size = 20, normalized size = 0.14 \[ -\frac{\, _2F_1\left (-\frac{1}{4},\frac{3}{2};\frac{3}{4};-x^4\right )}{x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 107, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{x}\sqrt{{x}^{4}+1}}+{\frac{{\frac{3\,i}{2}} \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^{2}}\,{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^{10} + 2 \, x^{6} + x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.937336, size = 31, normalized size = 0.22 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x \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}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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