Optimal. Leaf size=41 \[ \frac{5}{12} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )+\frac{5 x}{12 \sqrt{1-x^4}}+\frac{x}{6 \left (1-x^4\right )^{3/2}} \]
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Rubi [A] time = 0.0057407, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {199, 221} \[ \frac{5 x}{12 \sqrt{1-x^4}}+\frac{x}{6 \left (1-x^4\right )^{3/2}}+\frac{5}{12} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 199
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\left (1-x^4\right )^{5/2}} \, dx &=\frac{x}{6 \left (1-x^4\right )^{3/2}}+\frac{5}{6} \int \frac{1}{\left (1-x^4\right )^{3/2}} \, dx\\ &=\frac{x}{6 \left (1-x^4\right )^{3/2}}+\frac{5 x}{12 \sqrt{1-x^4}}+\frac{5}{12} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{x}{6 \left (1-x^4\right )^{3/2}}+\frac{5 x}{12 \sqrt{1-x^4}}+\frac{5}{12} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0164566, size = 51, normalized size = 1.24 \[ \frac{5}{12} x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )+\frac{5 x}{12 \sqrt{1-x^4}}+\frac{x}{6 \left (1-x^4\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 64, normalized size = 1.6 \begin{align*}{\frac{x}{6\, \left ({x}^{4}-1 \right ) ^{2}}\sqrt{-{x}^{4}+1}}+{\frac{5\,x}{12}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{12}\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{5}{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^{12} - 3 \, x^{8} + 3 \, x^{4} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.965805, size = 29, normalized size = 0.71 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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