Optimal. Leaf size=61 \[ i \text {Li}_2\left (\frac {2}{1-i x}-1\right )+\log \left (\frac {x^2}{x^2+1}\right ) \tan ^{-1}(x)+i \tan ^{-1}(x)^2-2 \log \left (2-\frac {2}{1-i x}\right ) \tan ^{-1}(x) \]
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Rubi [A] time = 0.11, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {203, 2526, 12, 4924, 4868, 2447} \[ i \text {PolyLog}\left (2,-1+\frac {2}{1-i x}\right )+\log \left (\frac {x^2}{x^2+1}\right ) \tan ^{-1}(x)+i \tan ^{-1}(x)^2-2 \log \left (2-\frac {2}{1-i x}\right ) \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 2447
Rule 2526
Rule 4868
Rule 4924
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {x^2}{1+x^2}\right )}{1+x^2} \, dx &=\tan ^{-1}(x) \log \left (\frac {x^2}{1+x^2}\right )-\int \frac {2 \tan ^{-1}(x)}{x \left (1+x^2\right )} \, dx\\ &=\tan ^{-1}(x) \log \left (\frac {x^2}{1+x^2}\right )-2 \int \frac {\tan ^{-1}(x)}{x \left (1+x^2\right )} \, dx\\ &=i \tan ^{-1}(x)^2+\tan ^{-1}(x) \log \left (\frac {x^2}{1+x^2}\right )-2 i \int \frac {\tan ^{-1}(x)}{x (i+x)} \, dx\\ &=i \tan ^{-1}(x)^2-2 \tan ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )+\tan ^{-1}(x) \log \left (\frac {x^2}{1+x^2}\right )+2 \int \frac {\log \left (2-\frac {2}{1-i x}\right )}{1+x^2} \, dx\\ &=i \tan ^{-1}(x)^2-2 \tan ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )+\tan ^{-1}(x) \log \left (\frac {x^2}{1+x^2}\right )+i \text {Li}_2\left (-1+\frac {2}{1-i x}\right )\\ \end {align*}
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Mathematica [B] time = 0.05, size = 239, normalized size = 3.92 \[ -\frac {1}{2} i \text {Li}_2\left (-\frac {1}{2} i (i-x)\right )+i \text {Li}_2(-i (i-x))+\frac {1}{2} i \text {Li}_2\left (-\frac {1}{2} i (x+i)\right )-i \text {Li}_2(-i (x+i))-\frac {1}{2} i \log \left (\frac {x^2}{x^2+1}\right ) \log (-x+i)+\frac {1}{2} i \log (x+i) \log \left (\frac {x^2}{x^2+1}\right )-\frac {1}{4} i \log ^2(-x+i)+\frac {1}{4} i \log ^2(x+i)+i \log (-i x) \log (-x+i)-\frac {1}{2} i \log \left (-\frac {1}{2} i (x+i)\right ) \log (-x+i)+\frac {1}{2} i \log \left (-\frac {1}{2} i (-x+i)\right ) \log (x+i)-i \log (i x) \log (x+i) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {x^{2}}{x^{2} + 1}\right )}{x^{2} + 1}, x\right ) \]
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 \[ \int \frac {\log \left (\frac {x^{2}}{x^{2} + 1}\right )}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 158, normalized size = 2.59 \[ i \ln \left (-i x \right ) \ln \left (x -i\right )-i \ln \left (i x \right ) \ln \left (x +i\right )-\frac {i \ln \left (-\frac {i \left (x +i\right )}{2}\right ) \ln \left (x -i\right )}{2}+\frac {i \ln \left (\frac {i \left (x -i\right )}{2}\right ) \ln \left (x +i\right )}{2}-\frac {i \ln \left (\frac {x^{2}}{x^{2}+1}\right ) \ln \left (x -i\right )}{2}+\frac {i \ln \left (\frac {x^{2}}{x^{2}+1}\right ) \ln \left (x +i\right )}{2}-\frac {i \ln \left (x -i\right )^{2}}{4}+\frac {i \ln \left (x +i\right )^{2}}{4}+i \dilog \left (-i x \right )-i \dilog \left (i x \right )-\frac {i \dilog \left (-\frac {i \left (x +i\right )}{2}\right )}{2}+\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {x^{2}}{x^{2} + 1}\right )}{x^{2} + 1}\,{d x} \]
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 \[ \int \frac {\ln \left (\frac {x^2}{x^2+1}\right )}{x^2+1} \,d x \]
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 \[ \int \frac {\log {\left (\frac {x^{2}}{x^{2} + 1} \right )}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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