Optimal. Leaf size=15 \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]
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Rubi [A] time = 0.007262, antiderivative size = 15, normalized size of antiderivative = 1., 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 = {266, 36, 31, 29} \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 31
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x \left (1-x^8\right )} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{(1-x) x} \, dx,x,x^8\right )\\ &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,x^8\right )+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^8\right )\\ &=\log (x)-\frac{1}{8} \log \left (1-x^8\right )\\ \end{align*}
Mathematica [A] time = 0.0033479, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 32, normalized size = 2.1 \begin{align*} -{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}+\ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ({x}^{4}+1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954012, size = 20, normalized size = 1.33 \begin{align*} -\frac{1}{8} \, \log \left (x^{8} - 1\right ) + \frac{1}{8} \, \log \left (x^{8}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26036, size = 38, normalized size = 2.53 \begin{align*} -\frac{1}{8} \, \log \left (x^{8} - 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111104, size = 10, normalized size = 0.67 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{8} - 1 \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16158, size = 22, normalized size = 1.47 \begin{align*} \frac{1}{8} \, \log \left (x^{8}\right ) - \frac{1}{8} \, \log \left ({\left | x^{8} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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