Optimal. Leaf size=37 \[ \frac {1}{4 \left (1-x^4\right )}-\frac {1}{4 x^4}-\frac {1}{2} \log \left (1-x^4\right )+2 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {28, 266, 44} \[ \frac {1}{4 \left (1-x^4\right )}-\frac {1}{4 x^4}-\frac {1}{2} \log \left (1-x^4\right )+2 \log (x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^5 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^2 x^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{(-1+x)^2}-\frac {2}{-1+x}+\frac {1}{x^2}+\frac {2}{x}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4 \left (1-x^4\right )}+2 \log (x)-\frac {1}{2} \log \left (1-x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.95 \[ -\frac {1}{4 \left (x^4-1\right )}-\frac {1}{4 x^4}-\frac {1}{2} \log \left (1-x^4\right )+2 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 50, normalized size = 1.35 \[ -\frac {2 \, x^{4} + 2 \, {\left (x^{8} - x^{4}\right )} \log \left (x^{4} - 1\right ) - 8 \, {\left (x^{8} - x^{4}\right )} \log \relax (x) - 1}{4 \, {\left (x^{8} - x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 36, normalized size = 0.97 \[ -\frac {2 \, x^{4} - 1}{4 \, {\left (x^{8} - x^{4}\right )}} + \frac {1}{2} \, \log \left (x^{4}\right ) - \frac {1}{2} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 54, normalized size = 1.46 \[ 2 \ln \relax (x )-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x^{2}+1\right )}{2}-\frac {1}{4 x^{4}}+\frac {1}{16 x +16}+\frac {1}{8 x^{2}+8}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 35, normalized size = 0.95 \[ -\frac {2 \, x^{4} - 1}{4 \, {\left (x^{8} - x^{4}\right )}} - \frac {1}{2} \, \log \left (x^{4} - 1\right ) + \frac {1}{2} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.86 \[ 2\,\ln \relax (x)-\frac {\ln \left (x^4-1\right )}{2}+\frac {\frac {x^4}{2}-\frac {1}{4}}{x^4-x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 29, normalized size = 0.78 \[ \frac {1 - 2 x^{4}}{4 x^{8} - 4 x^{4}} + 2 \log {\relax (x )} - \frac {\log {\left (x^{4} - 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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