Integrand size = 11, antiderivative size = 22 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=-\frac {1}{6 x^6}-\log (x)+\frac {1}{6} \log \left (1+x^6\right ) \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {272, 46} \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=-\frac {1}{6 x^6}+\frac {1}{6} \log \left (x^6+1\right )-\log (x) \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{x^2 (1+x)} \, dx,x,x^6\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,x^6\right ) \\ & = -\frac {1}{6 x^6}-\log (x)+\frac {1}{6} \log \left (1+x^6\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=-\frac {1}{6 x^6}-\log (x)+\frac {1}{6} \log \left (1+x^6\right ) \]
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Time = 4.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
meijerg | \(-\frac {1}{6 x^{6}}-\ln \left (x \right )+\frac {\ln \left (x^{6}+1\right )}{6}\) | \(19\) |
risch | \(-\frac {1}{6 x^{6}}-\ln \left (x \right )+\frac {\ln \left (x^{6}+1\right )}{6}\) | \(19\) |
default | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{6}-\frac {1}{6 x^{6}}-\ln \left (x \right )+\frac {\ln \left (x^{2}+1\right )}{6}\) | \(32\) |
norman | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{6}-\frac {1}{6 x^{6}}-\ln \left (x \right )+\frac {\ln \left (x^{2}+1\right )}{6}\) | \(32\) |
parallelrisch | \(-\frac {6 \ln \left (x \right ) x^{6}-\ln \left (x^{2}+1\right ) x^{6}-\ln \left (x^{4}-x^{2}+1\right ) x^{6}+1}{6 x^{6}}\) | \(42\) |
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none
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=\frac {x^{6} \log \left (x^{6} + 1\right ) - 6 \, x^{6} \log \left (x\right ) - 1}{6 \, x^{6}} \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=- \log {\left (x \right )} + \frac {\log {\left (x^{6} + 1 \right )}}{6} - \frac {1}{6 x^{6}} \]
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none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=-\frac {1}{6 \, x^{6}} + \frac {1}{6} \, \log \left (x^{6} + 1\right ) - \frac {1}{6} \, \log \left (x^{6}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=\frac {x^{6} - 1}{6 \, x^{6}} + \frac {1}{6} \, \log \left (x^{6} + 1\right ) - \frac {1}{6} \, \log \left (x^{6}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^7 \left (1+x^6\right )} \, dx=\frac {\ln \left (x^6+1\right )}{6}-\ln \left (x\right )-\frac {1}{6\,x^6} \]
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