Integrand size = 16, antiderivative size = 41 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{18 x^6}+\frac {4}{27 x^3}+\frac {13 \log (x)}{27}-\frac {1}{6} \log \left (1+x^3\right )+\frac {1}{162} \log \left (3+x^3\right ) \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1371, 723, 814} \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{18 x^6}+\frac {4}{27 x^3}-\frac {1}{6} \log \left (x^3+1\right )+\frac {1}{162} \log \left (x^3+3\right )+\frac {13 \log (x)}{27} \]
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Rule 723
Rule 814
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 \left (3+4 x+x^2\right )} \, dx,x,x^3\right ) \\ & = -\frac {1}{18 x^6}+\frac {1}{9} \text {Subst}\left (\int \frac {-4-x}{x^2 \left (3+4 x+x^2\right )} \, dx,x,x^3\right ) \\ & = -\frac {1}{18 x^6}+\frac {1}{9} \text {Subst}\left (\int \left (-\frac {4}{3 x^2}+\frac {13}{9 x}-\frac {3}{2 (1+x)}+\frac {1}{18 (3+x)}\right ) \, dx,x,x^3\right ) \\ & = -\frac {1}{18 x^6}+\frac {4}{27 x^3}+\frac {13 \log (x)}{27}-\frac {1}{6} \log \left (1+x^3\right )+\frac {1}{162} \log \left (3+x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{18 x^6}+\frac {4}{27 x^3}+\frac {13 \log (x)}{27}-\frac {1}{6} \log \left (1+x^3\right )+\frac {1}{162} \log \left (3+x^3\right ) \]
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Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {-\frac {1}{18}+\frac {4 x^{3}}{27}}{x^{6}}+\frac {13 \ln \left (x \right )}{27}-\frac {\ln \left (x^{3}+1\right )}{6}+\frac {\ln \left (x^{3}+3\right )}{162}\) | \(33\) |
default | \(-\frac {1}{18 x^{6}}+\frac {4}{27 x^{3}}+\frac {13 \ln \left (x \right )}{27}-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{3}+3\right )}{162}-\frac {\ln \left (x^{2}-x +1\right )}{6}\) | \(41\) |
norman | \(\frac {-\frac {1}{18}+\frac {4 x^{3}}{27}}{x^{6}}+\frac {13 \ln \left (x \right )}{27}-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{3}+3\right )}{162}-\frac {\ln \left (x^{2}-x +1\right )}{6}\) | \(42\) |
parallelrisch | \(\frac {78 \ln \left (x \right ) x^{6}-27 \ln \left (x +1\right ) x^{6}+\ln \left (x^{3}+3\right ) x^{6}-27 \ln \left (x^{2}-x +1\right ) x^{6}-9+24 x^{3}}{162 x^{6}}\) | \(53\) |
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Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=\frac {x^{6} \log \left (x^{3} + 3\right ) - 27 \, x^{6} \log \left (x^{3} + 1\right ) + 78 \, x^{6} \log \left (x\right ) + 24 \, x^{3} - 9}{162 \, x^{6}} \]
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Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=\frac {13 \log {\left (x \right )}}{27} - \frac {\log {\left (x^{3} + 1 \right )}}{6} + \frac {\log {\left (x^{3} + 3 \right )}}{162} + \frac {8 x^{3} - 3}{54 x^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=\frac {8 \, x^{3} - 3}{54 \, x^{6}} + \frac {1}{162} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) + \frac {13}{81} \, \log \left (x^{3}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=-\frac {13 \, x^{6} - 8 \, x^{3} + 3}{54 \, x^{6}} + \frac {1}{162} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) + \frac {13}{27} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^7 \left (3+4 x^3+x^6\right )} \, dx=\frac {\ln \left (x^3+3\right )}{162}-\frac {\ln \left (x^3+1\right )}{6}+\frac {13\,\ln \left (x\right )}{27}+\frac {\frac {4\,x^3}{27}-\frac {1}{18}}{x^6} \]
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