Optimal. Leaf size=35 \[ -\frac {1}{5} \sqrt {1-x^4} x^3-\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {321, 307, 221, 1181, 424} \[ -\frac {1}{5} \sqrt {1-x^4} x^3-\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 321
Rule 424
Rule 1181
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {1-x^4}} \, dx &=-\frac {1}{5} x^3 \sqrt {1-x^4}+\frac {3}{5} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {1-x^4}-\frac {3}{5} \int \frac {1}{\sqrt {1-x^4}} \, dx+\frac {3}{5} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {1-x^4}-\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {1-x^4}+\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.97 \[ \frac {1}{5} x^3 \left (\, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^4\right )-\sqrt {1-x^4}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1} x^{6}}{x^{4} - 1}, x\right ) \]
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 \[ \int \frac {x^{6}}{\sqrt {-x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.54 \[ -\frac {\sqrt {-x^{4}+1}\, x^{3}}{5}-\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (x , i\right )+\EllipticF \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}} \]
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 \[ \int \frac {x^{6}}{\sqrt {-x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^6}{\sqrt {1-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.55, size = 31, normalized size = 0.89 \[ \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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