Integrand size = 19, antiderivative size = 29 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=6 x+\frac {x^2}{2}+\frac {169}{4} \log (7-x)-\frac {1}{4} \log (1+x) \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1671, 646, 31} \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=\frac {x^2}{2}+6 x+\frac {169}{4} \log (7-x)-\frac {1}{4} \log (x+1) \]
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Rule 31
Rule 646
Rule 1671
Rubi steps \begin{align*} \text {integral}& = \int \left (6+x+\frac {2 (22+21 x)}{-7-6 x+x^2}\right ) \, dx \\ & = 6 x+\frac {x^2}{2}+2 \int \frac {22+21 x}{-7-6 x+x^2} \, dx \\ & = 6 x+\frac {x^2}{2}-\frac {1}{4} \int \frac {1}{1+x} \, dx+\frac {169}{4} \int \frac {1}{-7+x} \, dx \\ & = 6 x+\frac {x^2}{2}+\frac {169}{4} \log (7-x)-\frac {1}{4} \log (1+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=6 x+\frac {x^2}{2}+\frac {169}{4} \log (7-x)-\frac {1}{4} \log (1+x) \]
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Time = 0.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {x^{2}}{2}+6 x +\frac {169 \ln \left (x -7\right )}{4}-\frac {\ln \left (x +1\right )}{4}\) | \(22\) |
norman | \(\frac {x^{2}}{2}+6 x +\frac {169 \ln \left (x -7\right )}{4}-\frac {\ln \left (x +1\right )}{4}\) | \(22\) |
risch | \(\frac {x^{2}}{2}+6 x +\frac {169 \ln \left (x -7\right )}{4}-\frac {\ln \left (x +1\right )}{4}\) | \(22\) |
parallelrisch | \(\frac {x^{2}}{2}+6 x +\frac {169 \ln \left (x -7\right )}{4}-\frac {\ln \left (x +1\right )}{4}\) | \(22\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 6 \, x - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {169}{4} \, \log \left (x - 7\right ) \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=\frac {x^{2}}{2} + 6 x + \frac {169 \log {\left (x - 7 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} \]
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Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 6 \, x - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {169}{4} \, \log \left (x - 7\right ) \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 6 \, x - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {169}{4} \, \log \left ({\left | x - 7 \right |}\right ) \]
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Time = 9.81 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2-x+x^3}{-7-6 x+x^2} \, dx=6\,x-\frac {\ln \left (x+1\right )}{4}+\frac {169\,\ln \left (x-7\right )}{4}+\frac {x^2}{2} \]
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