Optimal. Leaf size=22 \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]
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Rubi [A] time = 0.0097526, antiderivative size = 22, normalized size of antiderivative = 1., 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 = {266, 44} \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^9 \left (1+x^8\right )} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,x^8\right )\\ &=\frac{1}{8} \operatorname{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*}
Mathematica [A] time = 0.0033918, size = 22, normalized size = 1. \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 19, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,{x}^{8}}}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{8}+1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980192, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{8 \, x^{8}} + \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 = 1.21145, size = 63, normalized size = 2.86 \begin{align*} \frac{x^{8} \log \left (x^{8} + 1\right ) - 8 \, x^{8} \log \left (x\right ) - 1}{8 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.153234, size = 17, normalized size = 0.77 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x^{8} + 1 \right )}}{8} - \frac{1}{8 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18975, size = 34, normalized size = 1.55 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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