Optimal. Leaf size=63 \[ \frac{15}{14} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{15 \sqrt{1-x^4}}{14 x^3}-\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}} \]
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Rubi [A] time = 0.0148437, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {290, 325, 221} \[ -\frac{15 \sqrt{1-x^4}}{14 x^3}-\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{x^8 \left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^7 \sqrt{1-x^4}}+\frac{9}{2} \int \frac{1}{x^8 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{45}{14} \int \frac{1}{x^4 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{9 \sqrt{1-x^4}}{14 x^7}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{9 \sqrt{1-x^4}}{14 x^7}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0030944, size = 20, normalized size = 0.32 \[ -\frac{\, _2F_1\left (-\frac{7}{4},\frac{3}{2};-\frac{3}{4};x^4\right )}{7 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 73, normalized size = 1.2 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{7\,{x}^{7}}\sqrt{-{x}^{4}+1}}-{\frac{4}{7\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x,i \right ) }{14}\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^{8}}\,{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^{16} - 2 \, x^{12} + x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.92494, size = 37, normalized size = 0.59 \begin{align*} \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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