Integrand size = 20, antiderivative size = 47 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=-\frac {\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )}+\frac {\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )} \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 36, 31} \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=\frac {\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )}-\frac {\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )} \]
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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}{\left (a^2+x\right ) \left (b^2+x\right )} \, dx,x,x^2\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{a^2+x} \, dx,x,x^2\right )}{2 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \frac {1}{b^2+x} \, dx,x,x^2\right )}{2 \left (a^2-b^2\right )} \\ & = -\frac {\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )}+\frac {\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=\frac {-\log \left (a^2+x^2\right )+\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )} \]
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Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(-\frac {\ln \left (a^{2}+x^{2}\right )-\ln \left (b^{2}+x^{2}\right )}{2 \left (a^{2}-b^{2}\right )}\) | \(33\) |
default | \(-\frac {\ln \left (a^{2}+x^{2}\right )}{2 \left (a^{2}-b^{2}\right )}+\frac {\ln \left (b^{2}+x^{2}\right )}{2 a^{2}-2 b^{2}}\) | \(44\) |
norman | \(-\frac {\ln \left (a^{2}+x^{2}\right )}{2 \left (a^{2}-b^{2}\right )}+\frac {\ln \left (b^{2}+x^{2}\right )}{2 a^{2}-2 b^{2}}\) | \(44\) |
risch | \(-\frac {\ln \left (-a^{2}-x^{2}\right )}{2 \left (a^{2}-b^{2}\right )}+\frac {\ln \left (b^{2}+x^{2}\right )}{2 a^{2}-2 b^{2}}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.68 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=-\frac {\log \left (a^{2} + x^{2}\right ) - \log \left (b^{2} + x^{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (36) = 72\).
Time = 0.35 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.57 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=\frac {\log {\left (- \frac {a^{4}}{2 \left (a - b\right ) \left (a + b\right )} + \frac {a^{2} b^{2}}{\left (a - b\right ) \left (a + b\right )} + \frac {a^{2}}{2} - \frac {b^{4}}{2 \left (a - b\right ) \left (a + b\right )} + \frac {b^{2}}{2} + x^{2} \right )}}{2 \left (a - b\right ) \left (a + b\right )} - \frac {\log {\left (\frac {a^{4}}{2 \left (a - b\right ) \left (a + b\right )} - \frac {a^{2} b^{2}}{\left (a - b\right ) \left (a + b\right )} + \frac {a^{2}}{2} + \frac {b^{4}}{2 \left (a - b\right ) \left (a + b\right )} + \frac {b^{2}}{2} + x^{2} \right )}}{2 \left (a - b\right ) \left (a + b\right )} \]
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Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=-\frac {\log \left (a^{2} + x^{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )}} + \frac {\log \left (b^{2} + x^{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=-\frac {\log \left (a^{2} + x^{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )}} + \frac {\log \left (b^{2} + x^{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 256, normalized size of antiderivative = 5.45 \[ \int \frac {x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx=\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x^2\,\left (8\,a^2+8\,b^2\right )+16\,a^2\,b^2}{2\,\left (a^2-b^2\right )}-4\,x^2\right )\,1{}\mathrm {i}}{2\,\left (a^2-b^2\right )}-\frac {\left (\frac {x^2\,\left (8\,a^2+8\,b^2\right )+16\,a^2\,b^2}{2\,\left (a^2-b^2\right )}+4\,x^2\right )\,1{}\mathrm {i}}{2\,\left (a^2-b^2\right )}}{\frac {\frac {x^2\,\left (8\,a^2+8\,b^2\right )+16\,a^2\,b^2}{2\,\left (a^2-b^2\right )}-4\,x^2}{2\,\left (a^2-b^2\right )}+\frac {\frac {x^2\,\left (8\,a^2+8\,b^2\right )+16\,a^2\,b^2}{2\,\left (a^2-b^2\right )}+4\,x^2}{2\,\left (a^2-b^2\right )}}\right )\,1{}\mathrm {i}}{a^2-b^2} \]
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