Optimal. Leaf size=42 \[ -\frac {1}{6} \left (1-x^4\right )^{3/2}+\sqrt {1-x^4}+\frac {1}{2 \sqrt {1-x^4}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, 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 = {266, 43} \[ -\frac {1}{6} \left (1-x^4\right )^{3/2}+\sqrt {1-x^4}+\frac {1}{2 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{(1-x)^{3/2}}-\frac {2}{\sqrt {1-x}}+\sqrt {1-x}\right ) \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1-x^4}}+\sqrt {1-x^4}-\frac {1}{6} \left (1-x^4\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.64 \[ \frac {-x^8-4 x^4+8}{6 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 28, normalized size = 0.67 \[ \frac {{\left (x^{8} + 4 \, x^{4} - 8\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 0.76 \[ -\frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} + \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.79 \[ \frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{8}+4 x^{4}-8\right )}{6 \left (-x^{4}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 32, normalized size = 0.76 \[ -\frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} + \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 25, normalized size = 0.60 \[ -\frac {{\left (x^4-1\right )}^2+6\,x^4-9}{6\,\sqrt {1-x^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.72, size = 39, normalized size = 0.93 \[ - \frac {x^{8}}{6 \sqrt {1 - x^{4}}} - \frac {2 x^{4}}{3 \sqrt {1 - x^{4}}} + \frac {4}{3 \sqrt {1 - x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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