Optimal. Leaf size=45 \[ -\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {1}{2 x^3 \sqrt {1-x^4}}+\frac {5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 221
Rule 290
Rule 325
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*}
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Mathematica [C] time = 0.00, 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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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1}}{x^{12} - 2 \, x^{8} + x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 59, normalized size = 1.31 \[ \frac {x}{2 \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{6 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^4\,{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.39, size = 34, normalized size = 0.76 \[ \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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