Optimal. Leaf size=55 \[ -\frac{4 \sqrt{1-x^4}}{15 x^2}-\frac{2 \sqrt{1-x^4}}{15 x^6}-\frac{\sqrt{1-x^4}}{10 x^{10}} \]
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Rubi [A] time = 0.0121978, antiderivative size = 55, 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 = {271, 264} \[ -\frac{4 \sqrt{1-x^4}}{15 x^2}-\frac{2 \sqrt{1-x^4}}{15 x^6}-\frac{\sqrt{1-x^4}}{10 x^{10}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^{11} \sqrt{1-x^4}} \, dx &=-\frac{\sqrt{1-x^4}}{10 x^{10}}+\frac{4}{5} \int \frac{1}{x^7 \sqrt{1-x^4}} \, dx\\ &=-\frac{\sqrt{1-x^4}}{10 x^{10}}-\frac{2 \sqrt{1-x^4}}{15 x^6}+\frac{8}{15} \int \frac{1}{x^3 \sqrt{1-x^4}} \, dx\\ &=-\frac{\sqrt{1-x^4}}{10 x^{10}}-\frac{2 \sqrt{1-x^4}}{15 x^6}-\frac{4 \sqrt{1-x^4}}{15 x^2}\\ \end{align*}
Mathematica [A] time = 0.0047617, size = 30, normalized size = 0.55 \[ -\frac{\sqrt{1-x^4} \left (8 x^8+4 x^4+3\right )}{30 x^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 38, normalized size = 0.7 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 8\,{x}^{8}+4\,{x}^{4}+3 \right ) }{30\,{x}^{10}}{\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 = 0.998039, size = 58, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}{3 \, x^{6}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{5}{2}}}{10 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56153, size = 63, normalized size = 1.15 \begin{align*} -\frac{{\left (8 \, x^{8} + 4 \, x^{4} + 3\right )} \sqrt{-x^{4} + 1}}{30 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.05081, size = 104, normalized size = 1.89 \begin{align*} \begin{cases} - \frac{4 \sqrt{-1 + \frac{1}{x^{4}}}}{15} - \frac{2 \sqrt{-1 + \frac{1}{x^{4}}}}{15 x^{4}} - \frac{\sqrt{-1 + \frac{1}{x^{4}}}}{10 x^{8}} & \text{for}\: \frac{1}{\left |{x^{4}}\right |} > 1 \\- \frac{4 i \sqrt{1 - \frac{1}{x^{4}}}}{15} - \frac{2 i \sqrt{1 - \frac{1}{x^{4}}}}{15 x^{4}} - \frac{i \sqrt{1 - \frac{1}{x^{4}}}}{10 x^{8}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18238, size = 38, normalized size = 0.69 \begin{align*} -\frac{1}{10} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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