Optimal. Leaf size=53 \[ \frac{3}{2} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{3 \sqrt{1-x^4}}{2 x}+\frac{1}{2 x \sqrt{1-x^4}}-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.0204178, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {290, 325, 307, 221, 1181, 424} \[ -\frac{3 \sqrt{1-x^4}}{2 x}+\frac{1}{2 x \sqrt{1-x^4}}+\frac{3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 307
Rule 221
Rule 1181
Rule 424
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}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{2} \int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{2 x \sqrt{1-x^4}}-\frac{3 \sqrt{1-x^4}}{2 x}-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0028323, size = 18, normalized size = 0.34 \[ -\frac{\, _2F_1\left (-\frac{1}{4},\frac{3}{2};\frac{3}{4};x^4\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 68, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{x}\sqrt{-{x}^{4}+1}}+{\frac{3\,{\it EllipticF} \left ( x,i \right ) -3\,{\it EllipticE} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\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 [A] time = 0.972371, size = 32, normalized size = 0.6 \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^{2 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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