Optimal. Leaf size=15 \[ \log (x)-\frac {1}{8} \log \left (1-x^8\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 31, 29}
\begin {gather*} \log (x)-\frac {1}{8} \log \left (1-x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (1-x^8\right )} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,x^8\right )\\ &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,x^8\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^8\right )\\ &=\log (x)-\frac {1}{8} \log \left (1-x^8\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \log (x)-\frac {1}{8} \log \left (1-x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs.
\(2(13)=26\).
time = 0.18, size = 32, normalized size = 2.13
method | result | size |
risch | \(\ln \left (x \right )-\frac {\ln \left (x^{8}-1\right )}{8}\) | \(12\) |
meijerg | \(-\frac {\ln \left (-x^{8}+1\right )}{8}+\ln \left (x \right )+\frac {i \pi }{8}\) | \(18\) |
default | \(-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8}-\frac {\ln \left (x -1\right )}{8}+\ln \left (x \right )-\frac {\ln \left (x^{2}+1\right )}{8}\) | \(32\) |
norman | \(-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8}-\frac {\ln \left (x -1\right )}{8}+\ln \left (x \right )-\frac {\ln \left (x^{2}+1\right )}{8}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \, \log \left (x^{8} - 1\right ) + \frac {1}{8} \, \log \left (x^{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{8} \, \log \left (x^{8} - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.67 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{8} - 1 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 16, normalized size = 1.07 \begin {gather*} \frac {1}{8} \, \log \left (x^{8}\right ) - \frac {1}{8} \, \log \left ({\left | x^{8} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.07, size = 11, normalized size = 0.73 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^8-1\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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