Optimal. Leaf size=45 \[ \frac{5}{6} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}} \]
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Rubi [A] time = 0.0097015, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {290, 325, 221} \[ -\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 221
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}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0026915, size = 20, normalized size = 0.44 \[ -\frac{\, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{1}{4};x^4\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 59, normalized size = 1.3 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{6}\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^{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 [A] time = 1.04721, size = 34, normalized size = 0.76 \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^{2 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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