Optimal. Leaf size=27 \[ \frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]
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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, 288, 216} \[ \frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 275
Rule 288
Rubi steps
\begin {align*} \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \sin ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.96 \[ \frac {1}{2} \left (\frac {x^2}{\sqrt {1-x^4}}-\sin ^{-1}\left (x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 46, normalized size = 1.70 \[ -\frac {\sqrt {-x^{4} + 1} x^{2} - 2 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right )}{2 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 28, normalized size = 1.04 \[ -\frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} - \frac {1}{2} \, \arcsin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 62, normalized size = 2.30 \[ -\frac {\arcsin \left (x^{2}\right )}{2}-\frac {\sqrt {-2 x^{2}-\left (x^{2}-1\right )^{2}+2}}{4 \left (x^{2}-1\right )}-\frac {\sqrt {2 x^{2}-\left (x^{2}+1\right )^{2}+2}}{4 \left (x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 31, normalized size = 1.15 \[ \frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \]
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 \[ \int \frac {x^5}{{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 46, normalized size = 1.70 \[ \begin {cases} - \frac {i x^{2}}{2 \sqrt {x^{4} - 1}} + \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {x^{2}}{2 \sqrt {1 - x^{4}}} - \frac {\operatorname {asin}{\left (x^{2} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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