Optimal. Leaf size=43 \[ -\frac {5}{21} x \sqrt {1-x^4}-\frac {1}{7} x^5 \sqrt {1-x^4}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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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 = {327, 227}
\begin {gather*} \frac {5}{21} F(\text {ArcSin}(x)|-1)-\frac {5}{21} \sqrt {1-x^4} x-\frac {1}{7} \sqrt {1-x^4} x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 327
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 42, normalized size = 0.98 \begin {gather*} \frac {1}{21} \left (-x \sqrt {1-x^4} \left (5+3 x^4\right )+5 x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 59, normalized size = 1.37
method | result | size |
meijerg | \(\frac {x^{9} \hypergeom \left (\left [\frac {1}{2}, \frac {9}{4}\right ], \left [\frac {13}{4}\right ], x^{4}\right )}{9}\) | \(15\) |
risch | \(\frac {x \left (3 x^{4}+5\right ) \left (x^{4}-1\right )}{21 \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}\) | \(57\) |
default | \(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}-\frac {5 x \sqrt {-x^{4}+1}}{21}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}\) | \(59\) |
elliptic | \(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}-\frac {5 x \sqrt {-x^{4}+1}}{21}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}\) | \(59\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 20, normalized size = 0.47 \begin {gather*} -\frac {1}{21} \, {\left (3 \, x^{5} + 5 \, x\right )} \sqrt {-x^{4} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 31, normalized size = 0.72 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^8}{\sqrt {1-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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