Integrand size = 15, antiderivative size = 27 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=\frac {\arctan (x)}{2}-\frac {1}{2} \log (1-x)+\frac {1}{4} \log \left (1+x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2083, 649, 209, 266} \[ \int \frac {1}{1-x+x^2-x^3} \, dx=\frac {\arctan (x)}{2}+\frac {1}{4} \log \left (x^2+1\right )-\frac {1}{2} \log (1-x) \]
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Rule 209
Rule 266
Rule 649
Rule 2083
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2 (-1+x)}+\frac {1+x}{2 \left (1+x^2\right )}\right ) \, dx \\ & = -\frac {1}{2} \log (1-x)+\frac {1}{2} \int \frac {1+x}{1+x^2} \, dx \\ & = -\frac {1}{2} \log (1-x)+\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\frac {1}{2} \int \frac {x}{1+x^2} \, dx \\ & = \frac {\arctan (x)}{2}-\frac {1}{2} \log (1-x)+\frac {1}{4} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=\frac {\arctan (x)}{2}-\frac {1}{2} \log (1-x)+\frac {1}{4} \log \left (1+x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\arctan \left (x \right )}{2}\) | \(20\) |
risch | \(-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\arctan \left (x \right )}{2}\) | \(20\) |
parallelrisch | \(-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (x -i\right )}{4}-\frac {i \ln \left (x -i\right )}{4}+\frac {\ln \left (i+x \right )}{4}+\frac {i \ln \left (i+x \right )}{4}\) | \(38\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=\frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=- \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x^{2} + 1 \right )}}{4} + \frac {\operatorname {atan}{\left (x \right )}}{2} \]
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=\frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=\frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {1}{1-x+x^2-x^3} \, dx=-\frac {\ln \left (x-1\right )}{2}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]
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