Optimal. Leaf size=37 \[ -\frac {1}{4 x^4}+\frac {1}{4 \left (1-x^4\right )}+2 \log (x)-\frac {1}{2} \log \left (1-x^4\right ) \]
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Rubi [A]
time = 0.01, 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, 272, 46}
\begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 46
Rule 272
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} \text {Subst}\left (\int \frac {1}{(-1+x)^2 x^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 54, normalized size = 1.46
method | result | size |
risch | \(\frac {\frac {1}{4}-\frac {x^{4}}{2}}{x^{4} \left (x^{4}-1\right )}+2 \ln \left (x \right )-\frac {\ln \left (x^{4}-1\right )}{2}\) | \(32\) |
norman | \(\frac {\frac {1}{4}-\frac {x^{4}}{2}}{x^{4} \left (x^{4}-1\right )}+2 \ln \left (x \right )-\frac {\ln \left (-1+x \right )}{2}-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(44\) |
default | \(-\frac {1}{16 \left (-1+x \right )}-\frac {\ln \left (-1+x \right )}{2}+\frac {1}{8 x^{2}+8}-\frac {\ln \left (x^{2}+1\right )}{2}-\frac {1}{4 x^{4}}+2 \ln \left (x \right )+\frac {1}{16+16 x}-\frac {\ln \left (1+x \right )}{2}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.95 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 50, normalized size = 1.35 \begin {gather*} -\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 \left (x\right ) - 1}{4 \, {\left (x^{8} - x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 0.78 \begin {gather*} \frac {1 - 2 x^{4}}{4 x^{8} - 4 x^{4}} + 2 \log {\left (x \right )} - \frac {\log {\left (x^{4} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.20, size = 36, normalized size = 0.97 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} 2\,\ln \left (x\right )-\frac {\ln \left (x^4-1\right )}{2}+\frac {\frac {x^4}{2}-\frac {1}{4}}{x^4-x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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