Optimal. Leaf size=43 \[ -\frac {5}{21} \sqrt {1-x^4} x-\frac {1}{7} \sqrt {1-x^4} x^5+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 43, 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 = {321, 221} \[ -\frac {1}{7} \sqrt {1-x^4} x^5-\frac {5}{21} \sqrt {1-x^4} x+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 321
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {1-x^4}} \, dx &=-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{7} \int \frac {x^4}{\sqrt {1-x^4}} \, dx\\ &=-\frac {5}{21} x \sqrt {1-x^4}-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{21} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=-\frac {5}{21} x \sqrt {1-x^4}-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.98 \[ \frac {1}{21} \left (5 x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )-x \sqrt {1-x^4} \left (3 x^4+5\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1} x^{8}}{x^{4} - 1}, x\right ) \]
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 \[ \int \frac {x^{8}}{\sqrt {-x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 1.37 \[ -\frac {\sqrt {-x^{4}+1}\, x^{5}}{7}-\frac {5 \sqrt {-x^{4}+1}\, x}{21}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}} \]
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 \[ \int \frac {x^{8}}{\sqrt {-x^{4} + 1}}\,{d x} \]
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 \[ \int \frac {x^8}{\sqrt {1-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.01, size = 31, normalized size = 0.72 \[ \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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