Integrand size = 16, antiderivative size = 35 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=-\frac {4 x^3}{3}+\frac {x^6}{6}-\frac {1}{6} \log \left (1+x^3\right )+\frac {9}{2} \log \left (3+x^3\right ) \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 715, 646, 31} \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=\frac {x^6}{6}-\frac {4 x^3}{3}-\frac {1}{6} \log \left (x^3+1\right )+\frac {9}{2} \log \left (x^3+3\right ) \]
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Rule 31
Rule 646
Rule 715
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x^3}{3+4 x+x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-4+x+\frac {12+13 x}{3+4 x+x^2}\right ) \, dx,x,x^3\right ) \\ & = -\frac {4 x^3}{3}+\frac {x^6}{6}+\frac {1}{3} \text {Subst}\left (\int \frac {12+13 x}{3+4 x+x^2} \, dx,x,x^3\right ) \\ & = -\frac {4 x^3}{3}+\frac {x^6}{6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^3\right )+\frac {9}{2} \text {Subst}\left (\int \frac {1}{3+x} \, dx,x,x^3\right ) \\ & = -\frac {4 x^3}{3}+\frac {x^6}{6}-\frac {1}{6} \log \left (1+x^3\right )+\frac {9}{2} \log \left (3+x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=-\frac {4 x^3}{3}+\frac {x^6}{6}-\frac {1}{6} \log \left (1+x^3\right )+\frac {9}{2} \log \left (3+x^3\right ) \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {4 x^{3}}{3}+\frac {x^{6}}{6}-\frac {\ln \left (x^{3}+1\right )}{6}+\frac {9 \ln \left (x^{3}+3\right )}{2}\) | \(28\) |
risch | \(\frac {x^{6}}{6}-\frac {4 x^{3}}{3}+\frac {8}{3}-\frac {\ln \left (x^{3}+1\right )}{6}+\frac {9 \ln \left (x^{3}+3\right )}{2}\) | \(29\) |
norman | \(-\frac {4 x^{3}}{3}+\frac {x^{6}}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {9 \ln \left (x^{3}+3\right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{6}\) | \(37\) |
parallelrisch | \(-\frac {4 x^{3}}{3}+\frac {x^{6}}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {9 \ln \left (x^{3}+3\right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{6}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=\frac {1}{6} \, x^{6} - \frac {4}{3} \, x^{3} + \frac {9}{2} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=\frac {x^{6}}{6} - \frac {4 x^{3}}{3} - \frac {\log {\left (x^{3} + 1 \right )}}{6} + \frac {9 \log {\left (x^{3} + 3 \right )}}{2} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=\frac {1}{6} \, x^{6} - \frac {4}{3} \, x^{3} + \frac {9}{2} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) \]
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Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=\frac {1}{6} \, x^{6} - \frac {4}{3} \, x^{3} + \frac {9}{2} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {x^{11}}{3+4 x^3+x^6} \, dx=\frac {9\,\ln \left (x^3+3\right )}{2}-\frac {\ln \left (x^3+1\right )}{6}-\frac {4\,x^3}{3}+\frac {x^6}{6} \]
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