Integrand size = 30, antiderivative size = 29 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=-3 \arctan (x)+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )+\frac {3}{2} \log \left (1+x^2\right ) \]
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Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6857, 649, 209, 266} \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=-3 \arctan (x)+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )+\frac {3}{2} \log \left (x^2+1\right ) \]
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Rule 209
Rule 266
Rule 649
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 (-1+x)}{1+x^2}+\frac {2}{2+x^2}\right ) \, dx \\ & = 2 \int \frac {1}{2+x^2} \, dx+3 \int \frac {-1+x}{1+x^2} \, dx \\ & = \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-3 \int \frac {1}{1+x^2} \, dx+3 \int \frac {x}{1+x^2} \, dx \\ & = -3 \tan ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )+\frac {3}{2} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=-3 \arctan (x)+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )+\frac {3}{2} \log \left (1+x^2\right ) \]
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Time = 0.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86
method | result | size |
default | \(-3 \arctan \left (x \right )+\frac {3 \ln \left (x^{2}+1\right )}{2}+\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}\) | \(25\) |
risch | \(-3 \arctan \left (x \right )+\frac {3 \ln \left (x^{2}+1\right )}{2}+\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}\) | \(25\) |
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 3 \, \arctan \left (x\right ) + \frac {3}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=\frac {3 \log {\left (x^{2} + 1 \right )}}{2} - 3 \operatorname {atan}{\left (x \right )} + \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 3 \, \arctan \left (x\right ) + \frac {3}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 3 \, \arctan \left (x\right ) + \frac {3}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 9.89 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-4+6 x-x^2+3 x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx=-\sqrt {2}\,\mathrm {atan}\left (\frac {24\,\sqrt {2}}{24\,x-64}+\frac {32\,\sqrt {2}\,x}{24\,x-64}\right )+\ln \left (x-\mathrm {i}\right )\,\left (\frac {3}{2}+\frac {3}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {3}{2}-\frac {3}{2}{}\mathrm {i}\right ) \]
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