Integrand size = 12, antiderivative size = 89 \[ \int \frac {\log (1+x)}{1+x^2} \, dx=-\frac {1}{2} i \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (i-x)\right ) \log (1+x)+\frac {1}{2} i \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (i+x)\right ) \log (1+x)-\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) (1+x)\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) (1+x)\right ) \]
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Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2456, 2441, 2440, 2438} \[ \int \frac {\log (1+x)}{1+x^2} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) (x+1)\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) (x+1)\right )-\frac {1}{2} i \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (-x+i)\right ) \log (x+1)+\frac {1}{2} i \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (x+i)\right ) \log (x+1) \]
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Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {i \log (1+x)}{2 (i-x)}+\frac {i \log (1+x)}{2 (i+x)}\right ) \, dx \\ & = \frac {1}{2} i \int \frac {\log (1+x)}{i-x} \, dx+\frac {1}{2} i \int \frac {\log (1+x)}{i+x} \, dx \\ & = -\frac {1}{2} i \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (i-x)\right ) \log (1+x)+\frac {1}{2} i \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (i+x)\right ) \log (1+x)+\frac {1}{2} i \int \frac {\log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (i-x)\right )}{1+x} \, dx-\frac {1}{2} i \int \frac {\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (i+x)\right )}{1+x} \, dx \\ & = -\frac {1}{2} i \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (i-x)\right ) \log (1+x)+\frac {1}{2} i \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (i+x)\right ) \log (1+x)-\frac {1}{2} i \text {Subst}\left (\int \frac {\log \left (1-\left (\frac {1}{2}+\frac {i}{2}\right ) x\right )}{x} \, dx,x,1+x\right )+\frac {1}{2} i \text {Subst}\left (\int \frac {\log \left (1-\left (\frac {1}{2}-\frac {i}{2}\right ) x\right )}{x} \, dx,x,1+x\right ) \\ & = -\frac {1}{2} i \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (i-x)\right ) \log (1+x)+\frac {1}{2} i \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (i+x)\right ) \log (1+x)-\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) (1+x)\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) (1+x)\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\log (1+x)}{1+x^2} \, dx=-\frac {1}{2} i \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (i-x)\right ) \log (1+x)+\frac {1}{2} i \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) (i+x)\right ) \log (1+x)-\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) (1+x)\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) (1+x)\right ) \]
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Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(-\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}\) | \(70\) |
default | \(-\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}\) | \(70\) |
risch | \(-\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}\) | \(70\) |
parts | \(-\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \ln \left (1+x \right ) \ln \left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}+\frac {i \left (1+x \right )}{2}\right )}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {x}{2}-\frac {i \left (1+x \right )}{2}\right )}{2}\) | \(70\) |
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\[ \int \frac {\log (1+x)}{1+x^2} \, dx=\int { \frac {\log \left (x + 1\right )}{x^{2} + 1} \,d x } \]
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\[ \int \frac {\log (1+x)}{1+x^2} \, dx=\int \frac {\log {\left (x + 1 \right )}}{x^{2} + 1}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63 \[ \int \frac {\log (1+x)}{1+x^2} \, dx=\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}, \frac {1}{2} \, x + \frac {1}{2}\right ) \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \arctan \left (x\right ) \log \left (\frac {1}{2} \, x^{2} + x + \frac {1}{2}\right ) + \arctan \left (x\right ) \log \left (x + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, x + \frac {1}{2} i + \frac {1}{2}\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, x - \frac {1}{2} i + \frac {1}{2}\right ) \]
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\[ \int \frac {\log (1+x)}{1+x^2} \, dx=\int { \frac {\log \left (x + 1\right )}{x^{2} + 1} \,d x } \]
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Timed out. \[ \int \frac {\log (1+x)}{1+x^2} \, dx=\int \frac {\ln \left (x+1\right )}{x^2+1} \,d x \]
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