Optimal. Leaf size=22 \[ -\frac {1}{8 x^8}+\log (x)-\frac {1}{8} \log \left (1-x^8\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46}
\begin {gather*} -\frac {1}{8 x^8}-\frac {1}{8} \log \left (1-x^8\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^9 \left (1-x^8\right )} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{(1-x) x^2} \, dx,x,x^8\right )\\ &=\frac {1}{8} \text {Subst}\left (\int \left (\frac {1}{1-x}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx,x,x^8\right )\\ &=-\frac {1}{8 x^8}+\log (x)-\frac {1}{8} \log \left (1-x^8\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{8 x^8}+\log (x)-\frac {1}{8} \log \left (1-x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 37, normalized size = 1.68
method | result | size |
risch | \(-\frac {1}{8 x^{8}}+\ln \left (x \right )-\frac {\ln \left (x^{8}-1\right )}{8}\) | \(17\) |
meijerg | \(-\frac {\ln \left (-x^{8}+1\right )}{8}+\ln \left (x \right )+\frac {i \pi }{8}-\frac {1}{8 x^{8}}\) | \(23\) |
default | \(-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8}-\frac {\ln \left (x -1\right )}{8}-\frac {1}{8 x^{8}}+\ln \left (x \right )-\frac {\ln \left (x^{2}+1\right )}{8}\) | \(37\) |
norman | \(-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8}-\frac {\ln \left (x -1\right )}{8}-\frac {1}{8 x^{8}}+\ln \left (x \right )-\frac {\ln \left (x^{2}+1\right )}{8}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{8 \, x^{8}} - \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 = 0.35, size = 24, normalized size = 1.09 \begin {gather*} -\frac {x^{8} \log \left (x^{8} - 1\right ) - 8 \, x^{8} \log \left (x\right ) + 1}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 17, normalized size = 0.77 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{8} - 1 \right )}}{8} - \frac {1}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.50, size = 26, normalized size = 1.18 \begin {gather*} -\frac {x^{8} + 1}{8 \, x^{8}} + \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 = 0.04, size = 16, normalized size = 0.73 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^8-1\right )}{8}-\frac {1}{8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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