Optimal. Leaf size=55 \[ -\frac {1}{6 \sqrt {1-x^4} x^6}+\frac {4 x^2}{3 \sqrt {1-x^4}}-\frac {2}{3 \sqrt {1-x^4} x^2} \]
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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} \[ \frac {4 x^2}{3 \sqrt {1-x^4}}-\frac {2}{3 \sqrt {1-x^4} x^2}-\frac {1}{6 \sqrt {1-x^4} x^6} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx &=-\frac {1}{6 x^6 \sqrt {1-x^4}}+\frac {4}{3} \int \frac {1}{x^3 \left (1-x^4\right )^{3/2}} \, dx\\ &=-\frac {1}{6 x^6 \sqrt {1-x^4}}-\frac {2}{3 x^2 \sqrt {1-x^4}}+\frac {8}{3} \int \frac {x}{\left (1-x^4\right )^{3/2}} \, dx\\ &=-\frac {1}{6 x^6 \sqrt {1-x^4}}-\frac {2}{3 x^2 \sqrt {1-x^4}}+\frac {4 x^2}{3 \sqrt {1-x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.55 \[ -\frac {-8 x^8+4 x^4+1}{6 x^6 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 34, normalized size = 0.62 \[ -\frac {{\left (8 \, x^{8} - 4 \, x^{4} - 1\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{10} - x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 94, normalized size = 1.71 \[ \frac {x^{6} {\left (\frac {21 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{2}}{x^{4}} + 1\right )}}{48 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}} - \frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} - \frac {7 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}}{16 \, x^{2}} - \frac {{\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}}{48 \, x^{6}} \]
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 \[ -\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (8 x^{8}-4 x^{4}-1\right )}{6 \left (-x^{4}+1\right )^{\frac {3}{2}} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 43, normalized size = 0.78 \[ \frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} - \frac {\sqrt {-x^{4} + 1}}{x^{2}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 30, normalized size = 0.55 \[ \frac {8\,{\left (x^4-1\right )}^2+12\,x^4-9}{6\,x^6\,\sqrt {1-x^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.36, size = 151, normalized size = 2.75 \[ \begin {cases} - \frac {8 x^{8} \sqrt {-1 + \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {4 x^{4} \sqrt {-1 + \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {\sqrt {-1 + \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {8 i x^{8} \sqrt {1 - \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {4 i x^{4} \sqrt {1 - \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {i \sqrt {1 - \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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