Optimal. Leaf size=27 \[ \frac {1}{4} \sin ^{-1}\left (x^2\right )-\frac {1}{4} x^2 \sqrt {1-x^4} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {275, 321, 216} \[ \frac {1}{4} \sin ^{-1}\left (x^2\right )-\frac {1}{4} x^2 \sqrt {1-x^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {1-x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{4} x^2 \sqrt {1-x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{4} x^2 \sqrt {1-x^4}+\frac {1}{4} \sin ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {1}{4} \sin ^{-1}\left (x^2\right )-\frac {1}{4} x^2 \sqrt {1-x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 33, normalized size = 1.22 \[ -\frac {1}{4} \, \sqrt {-x^{4} + 1} x^{2} - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 21, normalized size = 0.78 \[ -\frac {1}{4} \, \sqrt {-x^{4} + 1} x^{2} + \frac {1}{4} \, \arcsin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.81 \[ -\frac {\sqrt {-x^{4}+1}\, x^{2}}{4}+\frac {\arcsin \left (x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.97, size = 44, normalized size = 1.63 \[ \frac {\sqrt {-x^{4} + 1}}{4 \, x^{2} {\left (\frac {x^{4} - 1}{x^{4}} - 1\right )}} - \frac {1}{4} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^5}{\sqrt {1-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.18, size = 61, normalized size = 2.26 \[ \begin {cases} - \frac {i x^{2} \sqrt {x^{4} - 1}}{4} - \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{4} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {x^{6}}{4 \sqrt {1 - x^{4}}} - \frac {x^{2}}{4 \sqrt {1 - x^{4}}} + \frac {\operatorname {asin}{\left (x^{2} \right )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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