Optimal. Leaf size=45 \[ -\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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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 = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {325, 221} \[ -\frac {5 \sqrt {1-x^4}}{21 x^3}-\frac {\sqrt {1-x^4}}{7 x^7}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^8 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{7 x^7}+\frac {5}{7} \int \frac {1}{x^4 \sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3}+\frac {5}{21} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3}+\frac {5}{21} 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 \[ -\frac {\, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};x^4\right )}{7 x^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1}}{x^{12} - x^{8}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-x^{4} + 1} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 1.36 \[ \frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}-\frac {5 \sqrt {-x^{4}+1}}{21 x^{3}}-\frac {\sqrt {-x^{4}+1}}{7 x^{7}} \]
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 \[ \int \frac {1}{\sqrt {-x^{4} + 1} x^{8}}\,{d x} \]
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 \[ \int \frac {1}{x^8\,\sqrt {1-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.96, size = 37, normalized size = 0.82 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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