Optimal. Leaf size=45 \[ -\frac {3}{4} \sin ^{-1}\left (x^2\right )+\frac {x^6}{2 \sqrt {1-x^4}}+\frac {3}{4} \sqrt {1-x^4} x^2 \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {275, 288, 321, 216} \[ \frac {x^6}{2 \sqrt {1-x^4}}+\frac {3}{4} \sqrt {1-x^4} x^2-\frac {3}{4} \sin ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 275
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^9}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {x^6}{2 \sqrt {1-x^4}}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=\frac {x^6}{2 \sqrt {1-x^4}}+\frac {3}{4} x^2 \sqrt {1-x^4}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=\frac {x^6}{2 \sqrt {1-x^4}}+\frac {3}{4} x^2 \sqrt {1-x^4}-\frac {3}{4} \sin ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.91 \[ -\frac {x^6-3 x^2+3 \sqrt {1-x^4} \sin ^{-1}\left (x^2\right )}{4 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 52, normalized size = 1.16 \[ \frac {6 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right ) + {\left (x^{6} - 3 \, x^{2}\right )} \sqrt {-x^{4} + 1}}{4 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 33, normalized size = 0.73 \[ \frac {{\left (x^{4} - 3\right )} \sqrt {-x^{4} + 1} x^{2}}{4 \, {\left (x^{4} - 1\right )}} - \frac {3}{4} \, \arcsin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 76, normalized size = 1.69 \[ \frac {\sqrt {-x^{4}+1}\, x^{2}}{4}-\frac {3 \arcsin \left (x^{2}\right )}{4}-\frac {\sqrt {-2 x^{2}-\left (x^{2}-1\right )^{2}+2}}{4 \left (x^{2}-1\right )}-\frac {\sqrt {2 x^{2}-\left (x^{2}+1\right )^{2}+2}}{4 \left (x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 60, normalized size = 1.33 \[ -\frac {\frac {3 \, {\left (x^{4} - 1\right )}}{x^{4}} - 2}{4 \, {\left (\frac {\sqrt {-x^{4} + 1}}{x^{2}} + \frac {{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}{x^{6}}\right )}} + \frac {3}{4} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^9}{{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.16, size = 82, normalized size = 1.82 \[ \begin {cases} \frac {i x^{6}}{4 \sqrt {x^{4} - 1}} - \frac {3 i x^{2}}{4 \sqrt {x^{4} - 1}} + \frac {3 i \operatorname {acosh}{\left (x^{2} \right )}}{4} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {x^{6}}{4 \sqrt {1 - x^{4}}} + \frac {3 x^{2}}{4 \sqrt {1 - x^{4}}} - \frac {3 \operatorname {asin}{\left (x^{2} \right )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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