Optimal. Leaf size=57 \[ \frac {1}{2} i \log (x) \text {Li}_2(-i x)-\frac {1}{2} i \log (x) \text {Li}_2(i x)-\frac {1}{2} i \text {Li}_3(-i x)+\frac {1}{2} i \text {Li}_3(i x) \]
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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4940, 2438,
5125, 2421, 6724} \begin {gather*} -\frac {1}{2} i \text {Li}_3(-i x)+\frac {1}{2} i \text {Li}_3(i x)+\frac {1}{2} i \text {Li}_2(-i x) \log (x)-\frac {1}{2} i \text {Li}_2(i x) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2438
Rule 4940
Rule 5125
Rule 6724
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(x) \log (x)}{x} \, dx &=\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) \text {Li}_2(-i x)-\frac {1}{2} i \log (x) \text {Li}_2(i x)-\frac {1}{2} i \int \frac {\text {Li}_2(-i x)}{x} \, dx+\frac {1}{2} i \int \frac {\text {Li}_2(i x)}{x} \, dx\\ &=\frac {1}{2} i \log (x) \text {Li}_2(-i x)-\frac {1}{2} i \log (x) \text {Li}_2(i x)-\frac {1}{2} i \text {Li}_3(-i x)+\frac {1}{2} i \text {Li}_3(i x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 44, normalized size = 0.77 \begin {gather*} \frac {1}{2} i (\log (x) \text {Li}_2(-i x)-\log (x) \text {Li}_2(i x)-\text {Li}_3(-i x)+\text {Li}_3(i x)) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded in comparison} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.49, size = 71, normalized size = 1.25
method | result | size |
risch | \(\frac {i \ln \left (x \right )^{2} \ln \left (-i \left (x +i\right )\right )}{4}-\frac {i \ln \left (x \right )^{2} \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (x \right ) \polylog \left (2, i x \right )}{2}+\frac {i \polylog \left (3, i x \right )}{2}+\frac {i \ln \left (x \right ) \polylog \left (2, -i x \right )}{2}-\frac {i \polylog \left (3, -i x \right )}{2}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 31, normalized size = 0.54 \begin {gather*} -\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) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x \right )} \operatorname {atan}{\left (x \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {atan}\left (x\right )\,\ln \left (x\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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