Optimal. Leaf size=32 \[ \tan ^{-1}(x) \log (x)-\frac {1}{2} i \text {Li}_2(-i x)+\frac {1}{2} i \text {Li}_2(i x) \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {209, 2361,
4940, 2438} \begin {gather*} -\frac {1}{2} i \text {PolyLog}(2,-i x)+\frac {1}{2} i \text {PolyLog}(2,i x)+\text {ArcTan}(x) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2361
Rule 2438
Rule 4940
Rubi steps
\begin {align*} \int \frac {\log (x)}{1+x^2} \, dx &=\tan ^{-1}(x) \log (x)-\int \frac {\tan ^{-1}(x)}{x} \, dx\\ &=\tan ^{-1}(x) \log (x)-\frac {1}{2} i \int \frac {\log (1-i x)}{x} \, dx+\frac {1}{2} i \int \frac {\log (1+i x)}{x} \, dx\\ &=\tan ^{-1}(x) \log (x)-\frac {1}{2} i \text {Li}_2(-i x)+\frac {1}{2} i \text {Li}_2(i x)\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(65\) vs. \(2(32)=64\).
time = 0.01, size = 65, normalized size = 2.03 \begin {gather*} -\frac {1}{2} i \log (-i (i-x)) \log (x)+\frac {1}{2} i \log (x) \log (-i (i+x))-\frac {1}{2} i \text {Li}_2(-i x)+\frac {1}{2} i \text {Li}_2(i x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 46, normalized size = 1.44
method | result | size |
meijerg | \(\left (\frac {\ln \left (x \right ) \Phi \left (-x^{2}, 1, \frac {1}{2}\right )}{2}-\frac {\Phi \left (-x^{2}, 2, \frac {1}{2}\right )}{4}\right ) x\) | \(26\) |
default | \(-\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}-\frac {i \dilog \left (i x +1\right )}{2}+\frac {i \dilog \left (-i x +1\right )}{2}\) | \(46\) |
risch | \(-\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}-\frac {i \dilog \left (i x +1\right )}{2}+\frac {i \dilog \left (-i x +1\right )}{2}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, \pi \log \left (x^{2} + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (i \, x + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-i \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x \right )}}{x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 [B]
time = 3.33, size = 24, normalized size = 0.75 \begin {gather*} \mathrm {atan}\left (x\right )\,\ln \left (x\right )-\frac {\mathrm {polylog}\left (2,-x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {\mathrm {polylog}\left (2,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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