Optimal. Leaf size=13 \[ \log (x)-\frac {1}{8} \log \left (x^8+1\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {266, 36, 29, 31} \[ \log (x)-\frac {1}{8} \log \left (x^8+1\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+x^8\right )} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,x^8\right )\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^8\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{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 = 13, normalized size = 1.00 \[ \log (x)-\frac {1}{8} \log \left (x^8+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 11, normalized size = 0.85 \[ -\frac {1}{8} \, \log \left (x^{8} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 15, normalized size = 1.15 \[ -\frac {1}{8} \, \log \left (x^{8} + 1\right ) + \frac {1}{8} \, \log \left (x^{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.92 \[ \ln \relax (x )-\frac {\ln \left (x^{8}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 15, normalized size = 1.15 \[ -\frac {1}{8} \, \log \left (x^{8} + 1\right ) + \frac {1}{8} \, \log \left (x^{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 11, normalized size = 0.85 \[ \ln \relax (x)-\frac {\ln \left (x^8+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 10, normalized size = 0.77 \[ \log {\relax (x )} - \frac {\log {\left (x^{8} + 1 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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