Integrand size = 13, antiderivative size = 22 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=-\frac {1}{8 x^8}+\log (x)-\frac {1}{8} \log \left (1-x^8\right ) \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46} \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=-\frac {1}{8 x^8}-\frac {1}{8} \log \left (1-x^8\right )+\log (x) \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=-\frac {1}{8 x^8}+\log (x)-\frac {1}{8} \log \left (1-x^8\right ) \]
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Time = 3.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {1}{8 x^{8}}+\ln \left (x \right )-\frac {\ln \left (x^{8}-1\right )}{8}\) | \(17\) |
meijerg | \(-\frac {1}{8 x^{8}}+\ln \left (x \right )+\frac {i \pi }{8}-\frac {\ln \left (-x^{8}+1\right )}{8}\) | \(23\) |
default | \(-\frac {1}{8 x^{8}}+\ln \left (x \right )-\frac {\ln \left (-1+x \right )}{8}-\frac {\ln \left (1+x \right )}{8}-\frac {\ln \left (x^{2}+1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8}\) | \(37\) |
norman | \(-\frac {1}{8 x^{8}}+\ln \left (x \right )-\frac {\ln \left (-1+x \right )}{8}-\frac {\ln \left (1+x \right )}{8}-\frac {\ln \left (x^{2}+1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8}\) | \(37\) |
parallelrisch | \(\frac {8 \ln \left (x \right ) x^{8}-\ln \left (1+x \right ) x^{8}-\ln \left (-1+x \right ) x^{8}-\ln \left (x^{2}+1\right ) x^{8}-\ln \left (x^{4}+1\right ) x^{8}-1}{8 x^{8}}\) | \(55\) |
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=-\frac {x^{8} \log \left (x^{8} - 1\right ) - 8 \, x^{8} \log \left (x\right ) + 1}{8 \, x^{8}} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=\log {\left (x \right )} - \frac {\log {\left (x^{8} - 1 \right )}}{8} - \frac {1}{8 x^{8}} \]
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Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=-\frac {1}{8 \, x^{8}} - \frac {1}{8} \, \log \left (x^{8} - 1\right ) + \frac {1}{8} \, \log \left (x^{8}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=-\frac {x^{8} + 1}{8 \, x^{8}} + \frac {1}{8} \, \log \left (x^{8}\right ) - \frac {1}{8} \, \log \left ({\left | x^{8} - 1 \right |}\right ) \]
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Time = 5.87 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^9 \left (1-x^8\right )} \, dx=\ln \left (x\right )-\frac {\ln \left (x^8-1\right )}{8}-\frac {1}{8\,x^8} \]
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