Optimal. Leaf size=46 \[ -\frac{1}{10} \left (1-x^4\right )^{5/2}+\frac{1}{3} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]
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Rubi [A] time = 0.0201704, antiderivative size = 46, normalized size of antiderivative = 1., 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} \[ -\frac{1}{10} \left (1-x^4\right )^{5/2}+\frac{1}{3} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{1-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1-x}}-2 \sqrt{1-x}+(1-x)^{3/2}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{2} \sqrt{1-x^4}+\frac{1}{3} \left (1-x^4\right )^{3/2}-\frac{1}{10} \left (1-x^4\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0087716, size = 27, normalized size = 0.59 \[ -\frac{1}{30} \sqrt{1-x^4} \left (3 x^8+4 x^4+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.8 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 3\,{x}^{8}+4\,{x}^{4}+8 \right ) }{30}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997233, size = 46, normalized size = 1. \begin{align*} -\frac{1}{10} \,{\left (-x^{4} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45496, size = 57, normalized size = 1.24 \begin{align*} -\frac{1}{30} \,{\left (3 \, x^{8} + 4 \, x^{4} + 8\right )} \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.69958, size = 41, normalized size = 0.89 \begin{align*} - \frac{x^{8} \sqrt{1 - x^{4}}}{10} - \frac{2 x^{4} \sqrt{1 - x^{4}}}{15} - \frac{4 \sqrt{1 - x^{4}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12535, size = 55, normalized size = 1.2 \begin{align*} -\frac{1}{10} \,{\left (x^{4} - 1\right )}^{2} \sqrt{-x^{4} + 1} + \frac{1}{3} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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