Integrand size = 10, antiderivative size = 49 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {3 x}{2}-\frac {3 \arctan (x)}{2}-\frac {1}{2} x^2 \arctan (x)-\frac {1}{2} x \log \left (1+x^2\right )+\frac {1}{2} \left (1+x^2\right ) \arctan (x) \log \left (1+x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4946, 327, 209, 2504, 2436, 2332, 5139, 2498} \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=-\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} \left (x^2+1\right ) \arctan (x) \log \left (x^2+1\right )-\frac {3 \arctan (x)}{2}-\frac {1}{2} x \log \left (x^2+1\right )+\frac {3 x}{2} \]
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Rule 209
Rule 327
Rule 2332
Rule 2436
Rule 2498
Rule 2504
Rule 4946
Rule 5139
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} \left (1+x^2\right ) \arctan (x) \log \left (1+x^2\right )-\int \left (-\frac {x^2}{2 \left (1+x^2\right )}+\frac {1}{2} \log \left (1+x^2\right )\right ) \, dx \\ & = -\frac {1}{2} x^2 \arctan (x)+\frac {1}{2} \left (1+x^2\right ) \arctan (x) \log \left (1+x^2\right )+\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx-\frac {1}{2} \int \log \left (1+x^2\right ) \, dx \\ & = \frac {x}{2}-\frac {1}{2} x^2 \arctan (x)-\frac {1}{2} x \log \left (1+x^2\right )+\frac {1}{2} \left (1+x^2\right ) \arctan (x) \log \left (1+x^2\right )-\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\int \frac {x^2}{1+x^2} \, dx \\ & = \frac {3 x}{2}-\frac {\arctan (x)}{2}-\frac {1}{2} x^2 \arctan (x)-\frac {1}{2} x \log \left (1+x^2\right )+\frac {1}{2} \left (1+x^2\right ) \arctan (x) \log \left (1+x^2\right )-\int \frac {1}{1+x^2} \, dx \\ & = \frac {3 x}{2}-\frac {3 \arctan (x)}{2}-\frac {1}{2} x^2 \arctan (x)-\frac {1}{2} x \log \left (1+x^2\right )+\frac {1}{2} \left (1+x^2\right ) \arctan (x) \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {1}{2} \left (3 x-3 \arctan (x)-x^2 \arctan (x)+\left (-x+\left (1+x^2\right ) \arctan (x)\right ) \log \left (1+x^2\right )\right ) \]
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Time = 1.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98
method | result | size |
parallelrisch | \(\frac {x^{2} \arctan \left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {x^{2} \arctan \left (x \right )}{2}-\frac {x \ln \left (x^{2}+1\right )}{2}+\frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{2}+\frac {3 x}{2}-\frac {3 \arctan \left (x \right )}{2}\) | \(48\) |
default | \(\text {Expression too large to display}\) | \(2222\) |
risch | \(\text {Expression too large to display}\) | \(15978\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=-\frac {1}{2} \, {\left (x^{2} + 3\right )} \arctan \left (x\right ) + \frac {1}{2} \, {\left ({\left (x^{2} + 1\right )} \arctan \left (x\right ) - x\right )} \log \left (x^{2} + 1\right ) + \frac {3}{2} \, x \]
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Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.14 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {x^{2} \log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{2} - \frac {x^{2} \operatorname {atan}{\left (x \right )}}{2} - \frac {x \log {\left (x^{2} + 1 \right )}}{2} + \frac {3 x}{2} + \frac {\log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{2} - \frac {3 \operatorname {atan}{\left (x \right )}}{2} \]
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Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=-\frac {1}{2} \, {\left (x^{2} - {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 1\right )} \arctan \left (x\right ) - \frac {1}{2} \, x \log \left (x^{2} + 1\right ) + \frac {3}{2} \, x - \arctan \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (39) = 78\).
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.76 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {1}{4} \, \pi x^{2} \log \left (x^{2} + 1\right ) \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, x^{2} \arctan \left (\frac {1}{x}\right ) \log \left (x^{2} + 1\right ) - \frac {1}{4} \, \pi x^{2} \mathrm {sgn}\left (x\right ) + \frac {1}{2} \, x^{2} \arctan \left (\frac {1}{x}\right ) + \frac {1}{4} \, \pi \log \left (x^{2} + 1\right ) \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, x \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \arctan \left (\frac {1}{x}\right ) \log \left (x^{2} + 1\right ) + \frac {3}{2} \, x - \frac {3}{2} \, \arctan \left (x\right ) \]
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Time = 0.49 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int x \arctan (x) \log \left (1+x^2\right ) \, dx=\frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{2}-x\,\left (\frac {\ln \left (x^2+1\right )}{2}-\frac {3}{2}\right )-x^2\,\left (\frac {\mathrm {atan}\left (x\right )}{2}-\frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{2}\right )-\frac {3\,\mathrm {atan}\left (x\right )}{2} \]
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