Integrand size = 16, antiderivative size = 21 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {455, 36, 31} \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=\frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right ) \]
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Rule 31
Rule 36
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) (4+x)} \, dx,x,x^2\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{4+x} \, dx,x,x^2\right ) \\ & = \frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right ) \]
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Time = 2.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
norman | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
risch | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
parallelrisch | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=-\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=\frac {\log {\left (x^{2} + 1 \right )}}{6} - \frac {\log {\left (x^{2} + 4 \right )}}{6} \]
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none
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=-\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=-\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {3\,x^2}{5\,x^2+8}\right )}{3} \]
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