Optimal. Leaf size=39 \[ -\frac {5}{12 x^6}-\frac {5}{4 x^2}+\frac {5}{4} \tanh ^{-1}\left (x^2\right )+\frac {1}{4 x^6 \left (1-x^4\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {28, 275, 290, 325, 207} \[ \frac {1}{4 x^6 \left (1-x^4\right )}-\frac {5}{4 x^2}-\frac {5}{12 x^6}+\frac {5}{4} \tanh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 207
Rule 275
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^7 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (-1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{4 x^6 \left (1-x^4\right )}-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (-1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {5}{12 x^6}+\frac {1}{4 x^6 \left (1-x^4\right )}-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (-1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {5}{12 x^6}-\frac {5}{4 x^2}+\frac {1}{4 x^6 \left (1-x^4\right )}-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,x^2\right )\\ &=-\frac {5}{12 x^6}-\frac {5}{4 x^2}+\frac {1}{4 x^6 \left (1-x^4\right )}+\frac {5}{4} \tanh ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.26 \[ -\frac {1}{6 x^6}-\frac {1}{x^2}-\frac {5}{8} \log \left (1-x^2\right )+\frac {5}{8} \log \left (x^2+1\right )-\frac {x^2}{4 \left (x^4-1\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 59, normalized size = 1.51 \[ -\frac {30 \, x^{8} - 20 \, x^{4} - 15 \, {\left (x^{10} - x^{6}\right )} \log \left (x^{2} + 1\right ) + 15 \, {\left (x^{10} - x^{6}\right )} \log \left (x^{2} - 1\right ) - 4}{24 \, {\left (x^{10} - x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 42, normalized size = 1.08 \[ -\frac {x^{2}}{4 \, {\left (x^{4} - 1\right )}} - \frac {6 \, x^{4} + 1}{6 \, x^{6}} + \frac {5}{8} \, \log \left (x^{2} + 1\right ) - \frac {5}{8} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 55, normalized size = 1.41 \[ -\frac {5 \ln \left (x -1\right )}{8}-\frac {5 \ln \left (x +1\right )}{8}+\frac {5 \ln \left (x^{2}+1\right )}{8}-\frac {1}{x^{2}}-\frac {1}{6 x^{6}}+\frac {1}{16 x +16}-\frac {1}{8 \left (x^{2}+1\right )}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 42, normalized size = 1.08 \[ -\frac {15 \, x^{8} - 10 \, x^{4} - 2}{12 \, {\left (x^{10} - x^{6}\right )}} + \frac {5}{8} \, \log \left (x^{2} + 1\right ) - \frac {5}{8} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.82 \[ \frac {5\,\mathrm {atanh}\left (x^2\right )}{4}-\frac {-\frac {5\,x^8}{4}+\frac {5\,x^4}{6}+\frac {1}{6}}{x^6-x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 41, normalized size = 1.05 \[ - \frac {5 \log {\left (x^{2} - 1 \right )}}{8} + \frac {5 \log {\left (x^{2} + 1 \right )}}{8} + \frac {- 15 x^{8} + 10 x^{4} + 2}{12 x^{10} - 12 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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