Optimal. Leaf size=43 \[ -\frac {9}{20 x^5}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4 x}-\frac {9}{8} \tan ^{-1}(x)+\frac {9}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {28, 290, 325, 298, 203, 206} \[ \frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{20 x^5}-\frac {9}{4 x}-\frac {9}{8} \tan ^{-1}(x)+\frac {9}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 206
Rule 290
Rule 298
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^6 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4} \int \frac {1}{x^6 \left (-1+x^4\right )} \, dx\\ &=-\frac {9}{20 x^5}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4} \int \frac {1}{x^2 \left (-1+x^4\right )} \, dx\\ &=-\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4} \int \frac {x^2}{-1+x^4} \, dx\\ &=-\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}+\frac {9}{8} \int \frac {1}{1-x^2} \, dx-\frac {9}{8} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{8} \tan ^{-1}(x)+\frac {9}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.19 \[ -\frac {1}{5 x^5}-\frac {x^3}{4 \left (x^4-1\right )}-\frac {2}{x}-\frac {9}{16} \log (1-x)+\frac {9}{16} \log (x+1)-\frac {9}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 68, normalized size = 1.58 \[ -\frac {180 \, x^{8} - 144 \, x^{4} + 90 \, {\left (x^{9} - x^{5}\right )} \arctan \relax (x) - 45 \, {\left (x^{9} - x^{5}\right )} \log \left (x + 1\right ) + 45 \, {\left (x^{9} - x^{5}\right )} \log \left (x - 1\right ) - 16}{80 \, {\left (x^{9} - x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 43, normalized size = 1.00 \[ -\frac {x^{3}}{4 \, {\left (x^{4} - 1\right )}} - \frac {10 \, x^{4} + 1}{5 \, x^{5}} - \frac {9}{8} \, \arctan \relax (x) + \frac {9}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {9}{16} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 1.21 \[ -\frac {x}{8 \left (x^{2}+1\right )}-\frac {9 \arctan \relax (x )}{8}-\frac {9 \ln \left (x -1\right )}{16}+\frac {9 \ln \left (x +1\right )}{16}-\frac {2}{x}-\frac {1}{5 x^{5}}-\frac {1}{16 \left (x +1\right )}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 42, normalized size = 0.98 \[ -\frac {45 \, x^{8} - 36 \, x^{4} - 4}{20 \, {\left (x^{9} - x^{5}\right )}} - \frac {9}{8} \, \arctan \relax (x) + \frac {9}{16} \, \log \left (x + 1\right ) - \frac {9}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 34, normalized size = 0.79 \[ \frac {9\,\mathrm {atanh}\relax (x)}{8}-\frac {9\,\mathrm {atan}\relax (x)}{8}-\frac {-\frac {9\,x^8}{4}+\frac {9\,x^4}{5}+\frac {1}{5}}{x^5-x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 44, normalized size = 1.02 \[ - \frac {9 \log {\left (x - 1 \right )}}{16} + \frac {9 \log {\left (x + 1 \right )}}{16} - \frac {9 \operatorname {atan}{\relax (x )}}{8} + \frac {- 45 x^{8} + 36 x^{4} + 4}{20 x^{9} - 20 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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