Integrand size = 23, antiderivative size = 33 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=\frac {2-x}{2 \left (1+x^2\right )}+\frac {3 \arctan (x)}{2}-\frac {1}{2} \log \left (1+x^2\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1828, 649, 209, 266} \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=\frac {3 \arctan (x)}{2}+\frac {2-x}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right ) \]
[In]
[Out]
Rule 209
Rule 266
Rule 649
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \frac {2-x}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {-3+2 x}{1+x^2} \, dx \\ & = \frac {2-x}{2 \left (1+x^2\right )}+\frac {3}{2} \int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx \\ & = \frac {2-x}{2 \left (1+x^2\right )}+\frac {3}{2} \tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {2-x}{1+x^2}+3 \arctan (x)-\log \left (1+x^2\right )\right ) \]
[In]
[Out]
Time = 0.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {1-\frac {x}{2}}{x^{2}+1}-\frac {\ln \left (x^{2}+1\right )}{2}+\frac {3 \arctan \left (x \right )}{2}\) | \(27\) |
default | \(-\frac {\frac {x}{2}-1}{x^{2}+1}-\frac {\ln \left (x^{2}+1\right )}{2}+\frac {3 \arctan \left (x \right )}{2}\) | \(28\) |
meijerg | \(-\frac {x^{2}}{x^{2}+1}-\frac {\ln \left (x^{2}+1\right )}{2}-\frac {x}{x^{2}+1}+\frac {3 \arctan \left (x \right )}{2}+\frac {x}{2 x^{2}+2}\) | \(47\) |
parallelrisch | \(-\frac {3 i \ln \left (x -i\right ) x^{2}-3 i \ln \left (x +i\right ) x^{2}+2 \ln \left (x -i\right ) x^{2}+2 \ln \left (x +i\right ) x^{2}-4+3 i \ln \left (x -i\right )-3 i \ln \left (x +i\right )+2 \ln \left (x -i\right )+2 \ln \left (x +i\right )+2 x}{4 \left (x^{2}+1\right )}\) | \(87\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=\frac {3 \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) - {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x + 2}{2 \, {\left (x^{2} + 1\right )}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=- \frac {x - 2}{2 x^{2} + 2} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {3 \operatorname {atan}{\left (x \right )}}{2} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=-\frac {x - 2}{2 \, {\left (x^{2} + 1\right )}} + \frac {3}{2} \, \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=-\frac {x - 2}{2 \, {\left (x^{2} + 1\right )}} + \frac {3}{2} \, \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx=\frac {3\,\mathrm {atan}\left (x\right )}{2}-\frac {\ln \left (x^2+1\right )}{2}-\frac {x}{2\,\left (x^2+1\right )}+\frac {1}{x^2+1} \]
[In]
[Out]