Optimal. Leaf size=55 \[ -\frac {\sqrt {1-x^4}}{10 x^{10}}-\frac {2 \sqrt {1-x^4}}{15 x^6}-\frac {4 \sqrt {1-x^4}}{15 x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 55, 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 = {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 264
Rule 271
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.85, size = 26, normalized size = 0.47 \[ -\frac {{\left (8 \, x^{8} + 4 \, x^{4} + 3\right )} \sqrt {-x^{4} + 1}}{30 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 109, normalized size = 1.98 \[ \frac {x^{10} {\left (\frac {25 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{2}}{x^{4}} + \frac {150 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{4}}{x^{8}} + 3\right )}}{960 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{5}} - \frac {5 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}}{32 \, x^{2}} - \frac {5 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}}{192 \, x^{6}} - \frac {{\left (\sqrt {-x^{4} + 1} - 1\right )}^{5}}{320 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.69 \[ \frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (8 x^{8}+4 x^{4}+3\right )}{30 \sqrt {-x^{4}+1}\, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 43, normalized size = 0.78 \[ -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{6}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {5}{2}}}{10 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 26, normalized size = 0.47 \[ -\frac {\sqrt {1-x^4}\,\left (8\,x^8+4\,x^4+3\right )}{30\,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.69, size = 104, normalized size = 1.89 \[ \begin {cases} - \frac {4 \sqrt {-1 + \frac {1}{x^{4}}}}{15} - \frac {2 \sqrt {-1 + \frac {1}{x^{4}}}}{15 x^{4}} - \frac {\sqrt {-1 + \frac {1}{x^{4}}}}{10 x^{8}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {4 i \sqrt {1 - \frac {1}{x^{4}}}}{15} - \frac {2 i \sqrt {1 - \frac {1}{x^{4}}}}{15 x^{4}} - \frac {i \sqrt {1 - \frac {1}{x^{4}}}}{10 x^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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