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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {272, 46}
\begin {gather*} -\frac {1}{8 x^8}+\frac {1}{8} \log \left (x^8+1\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}{x^2 (1+x)} \, dx,x,x^8\right )\\ &=\frac {1}{8} \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{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 = 0.18, size = 19, normalized size = 0.86
method | result | size |
default | \(-\frac {1}{8 x^{8}}-\ln \left (x \right )+\frac {\ln \left (x^{8}+1\right )}{8}\) | \(19\) |
norman | \(-\frac {1}{8 x^{8}}-\ln \left (x \right )+\frac {\ln \left (x^{8}+1\right )}{8}\) | \(19\) |
meijerg | \(-\frac {1}{8 x^{8}}-\ln \left (x \right )+\frac {\ln \left (x^{8}+1\right )}{8}\) | \(19\) |
risch | \(-\frac {1}{8 x^{8}}-\ln \left (x \right )+\frac {\ln \left (x^{8}+1\right )}{8}\) | \(19\) |
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 = 0.36, 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 = 0.79, size = 25, normalized size = 1.14 \begin {gather*} \frac {x^{8} - 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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Mupad [B]
time = 0.04, size = 18, normalized size = 0.82 \begin {gather*} \frac {\ln \left (x^8+1\right )}{8}-\ln \left (x\right )-\frac {1}{8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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