Integrand size = 8, antiderivative size = 57 \[ \int \frac {\arctan (x) \log (x)}{x} \, dx=\frac {1}{2} i \log (x) \operatorname {PolyLog}(2,-i x)-\frac {1}{2} i \log (x) \operatorname {PolyLog}(2,i x)-\frac {1}{2} i \operatorname {PolyLog}(3,-i x)+\frac {1}{2} i \operatorname {PolyLog}(3,i x) \]
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Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4940, 2438, 5125, 2421, 6724} \[ \int \frac {\arctan (x) \log (x)}{x} \, dx=-\frac {1}{2} i \operatorname {PolyLog}(3,-i x)+\frac {1}{2} i \operatorname {PolyLog}(3,i x)+\frac {1}{2} i \operatorname {PolyLog}(2,-i x) \log (x)-\frac {1}{2} i \operatorname {PolyLog}(2,i x) \log (x) \]
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Rule 2421
Rule 2438
Rule 4940
Rule 5125
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} i \int \frac {\log (1-i x) \log (x)}{x} \, dx-\frac {1}{2} i \int \frac {\log (1+i x) \log (x)}{x} \, dx \\ & = \frac {1}{2} i \log (x) \operatorname {PolyLog}(2,-i x)-\frac {1}{2} i \log (x) \operatorname {PolyLog}(2,i x)-\frac {1}{2} i \int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx+\frac {1}{2} i \int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx \\ & = \frac {1}{2} i \log (x) \operatorname {PolyLog}(2,-i x)-\frac {1}{2} i \log (x) \operatorname {PolyLog}(2,i x)-\frac {1}{2} i \operatorname {PolyLog}(3,-i x)+\frac {1}{2} i \operatorname {PolyLog}(3,i x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.77 \[ \int \frac {\arctan (x) \log (x)}{x} \, dx=\frac {1}{2} i (\log (x) \operatorname {PolyLog}(2,-i x)-\log (x) \operatorname {PolyLog}(2,i x)-\operatorname {PolyLog}(3,-i x)+\operatorname {PolyLog}(3,i x)) \]
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Time = 0.91 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.25
method | result | size |
risch | \(\frac {i \ln \left (x \right )^{2} \ln \left (-i \left (x +i\right )\right )}{4}-\frac {i \ln \left (x \right )^{2} \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (x \right ) \operatorname {Li}_{2}\left (i x \right )}{2}+\frac {i \operatorname {Li}_{3}\left (i x \right )}{2}+\frac {i \ln \left (x \right ) \operatorname {Li}_{2}\left (-i x \right )}{2}-\frac {i \operatorname {Li}_{3}\left (-i x \right )}{2}\) | \(71\) |
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\[ \int \frac {\arctan (x) \log (x)}{x} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x\right )}{x} \,d x } \]
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\[ \int \frac {\arctan (x) \log (x)}{x} \, dx=\int \frac {\log {\left (x \right )} \operatorname {atan}{\left (x \right )}}{x}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.54 \[ \int \frac {\arctan (x) \log (x)}{x} \, dx=-\frac {1}{2} i \, {\rm Li}_2\left (i \, x\right ) \log \left (x\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-i \, x\right ) \log \left (x\right ) + \frac {1}{2} i \, {\rm Li}_{3}(i \, x) - \frac {1}{2} i \, {\rm Li}_{3}(-i \, x) \]
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\[ \int \frac {\arctan (x) \log (x)}{x} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\arctan (x) \log (x)}{x} \, dx=\int \frac {\mathrm {atan}\left (x\right )\,\ln \left (x\right )}{x} \,d x \]
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