Optimal. Leaf size=41 \[ \frac {5 x}{12 \sqrt {1-x^4}}+\frac {x}{6 \left (1-x^4\right )^{3/2}}+\frac {5}{12} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.02, 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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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1}}{x^{12} - 3 \, x^{8} + 3 \, x^{4} - 1}, x\right ) \]
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 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 64, normalized size = 1.56 \[ \frac {\sqrt {-x^{4}+1}\, x}{6 \left (x^{4}-1\right )^{2}}+\frac {5 x}{12 \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{12 \sqrt {-x^{4}+1}} \]
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 \[ \int \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 10, normalized size = 0.24 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {5}{2};\ \frac {5}{4};\ x^4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 29, normalized size = 0.71 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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