Optimal. Leaf size=61 \[ i \text{PolyLog}\left (2,-1+\frac{2}{1-i x}\right )+\log \left (\frac{x^2}{x^2+1}\right ) \tan ^{-1}(x)+i \tan ^{-1}(x)^2-2 \log \left (2-\frac{2}{1-i x}\right ) \tan ^{-1}(x) \]
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Rubi [A] time = 0.1055, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 203
Rule 2526
Rule 12
Rule 4924
Rule 4868
Rule 2447
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*}
Mathematica [B] time = 0.0442744, size = 239, normalized size = 3.92 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-\frac{1}{2} i (-x+i)\right )+i \text{PolyLog}(2,-i (-x+i))+\frac{1}{2} i \text{PolyLog}\left (2,-\frac{1}{2} i (x+i)\right )-i \text{PolyLog}(2,-i (x+i))-\frac{1}{2} i \log \left (\frac{x^2}{x^2+1}\right ) \log (-x+i)+\frac{1}{2} i \log (x+i) \log \left (\frac{x^2}{x^2+1}\right )-\frac{1}{4} i \log ^2(-x+i)+\frac{1}{4} i \log ^2(x+i)+i \log (-i x) \log (-x+i)-\frac{1}{2} i \log \left (-\frac{1}{2} i (x+i)\right ) \log (-x+i)+\frac{1}{2} i \log \left (-\frac{1}{2} i (-x+i)\right ) \log (x+i)-i \log (i x) \log (x+i) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 158, normalized size = 2.6 \begin{align*} -{\frac{i}{2}}\ln \left ({\frac{{x}^{2}}{{x}^{2}+1}} \right ) \ln \left ( x-i \right ) -{\frac{i}{4}} \left ( \ln \left ( x-i \right ) \right ) ^{2}-{\frac{i}{2}}\ln \left ( x-i \right ) \ln \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +i\ln \left ( x-i \right ) \ln \left ( -ix \right ) -{\frac{i}{2}}{\it dilog} \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +i{\it dilog} \left ( -ix \right ) +{\frac{i}{2}}\ln \left ({\frac{{x}^{2}}{{x}^{2}+1}} \right ) \ln \left ( x+i \right ) +{\frac{i}{4}} \left ( \ln \left ( x+i \right ) \right ) ^{2}+{\frac{i}{2}}\ln \left ( x+i \right ) \ln \left ({\frac{i}{2}} \left ( x-i \right ) \right ) -i\ln \left ( x+i \right ) \ln \left ( ix \right ) +{\frac{i}{2}}{\it dilog} \left ({\frac{i}{2}} \left ( x-i \right ) \right ) -i{\it dilog} \left ( ix \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{x^{2}}{x^{2} + 1}\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{x^{2}}{x^{2} + 1}\right )}{x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\frac{x^{2}}{x^{2} + 1} \right )}}{x^{2} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{x^{2}}{x^{2} + 1}\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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