Optimal. Leaf size=49 \[ \frac{1}{2} i \text{PolyLog}\left (2,-1+\frac{2}{1-i x}\right )+\frac{1}{2} i \cot ^{-1}(x)^2+\log \left (2-\frac{2}{1-i x}\right ) \cot ^{-1}(x) \]
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Rubi [A] time = 0.0729478, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4925, 4869, 2447} \[ \frac{1}{2} i \text{PolyLog}\left (2,-1+\frac{2}{1-i x}\right )+\frac{1}{2} i \cot ^{-1}(x)^2+\log \left (2-\frac{2}{1-i x}\right ) \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4925
Rule 4869
Rule 2447
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx &=\frac{1}{2} i \cot ^{-1}(x)^2+i \int \frac{\cot ^{-1}(x)}{x (i+x)} \, dx\\ &=\frac{1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (2-\frac{2}{1-i x}\right )+\int \frac{\log \left (2-\frac{2}{1-i x}\right )}{1+x^2} \, dx\\ &=\frac{1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (2-\frac{2}{1-i x}\right )+\frac{1}{2} i \text{Li}_2\left (-1+\frac{2}{1-i x}\right )\\ \end{align*}
Mathematica [B] time = 0.0596438, size = 251, normalized size = 5.12 \[ -\frac{1}{4} i \text{PolyLog}\left (2,-\frac{1}{2} i (-x+i)\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{x}\right )+\frac{1}{4} i \text{PolyLog}\left (2,-\frac{1}{2} i (x+i)\right )+\frac{1}{8} i \log ^2(-x+i)-\frac{1}{8} i \log ^2(x+i)-\frac{1}{4} i \log \left (-\frac{-x+i}{x}\right ) \log (-x+i)-\frac{1}{4} i \log \left (-\frac{1}{2} i (x+i)\right ) \log (-x+i)+\frac{1}{4} i \log \left (\frac{x+i}{x}\right ) \log (-x+i)+\frac{1}{4} i \log \left (-\frac{1}{2} i (-x+i)\right ) \log (x+i)-\frac{1}{4} i \log \left (-\frac{-x+i}{x}\right ) \log (x+i)+\frac{1}{4} i \log (x+i) \log \left (\frac{x+i}{x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.115, size = 163, normalized size = 3.3 \begin{align*} -{\frac{{\rm arccot} \left (x\right )\ln \left ({x}^{2}+1 \right ) }{2}}+{\rm arccot} \left (x\right )\ln \left ( x \right ) -{\frac{i}{2}}\ln \left ( x \right ) \ln \left ( 1+ix \right ) +{\frac{i}{2}}\ln \left ( x \right ) \ln \left ( 1-ix \right ) -{\frac{i}{2}}{\it dilog} \left ( 1+ix \right ) +{\frac{i}{2}}{\it dilog} \left ( 1-ix \right ) -{\frac{i}{8}} \left ( \ln \left ( x-i \right ) \right ) ^{2}-{\frac{i}{4}}\ln \left ( x-i \right ) \ln \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +{\frac{i}{4}}\ln \left ( x-i \right ) \ln \left ({x}^{2}+1 \right ) -{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +{\frac{i}{8}} \left ( \ln \left ( x+i \right ) \right ) ^{2}+{\frac{i}{4}}\ln \left ( x+i \right ) \ln \left ({\frac{i}{2}} \left ( x-i \right ) \right ) -{\frac{i}{4}}\ln \left ( x+i \right ) \ln \left ({x}^{2}+1 \right ) +{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( x-i \right ) \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{\operatorname{arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x}\,{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{\operatorname{arccot}\left (x\right )}{x^{3} + x}, 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{\operatorname{acot}{\left (x \right )}}{x \left (x^{2} + 1\right )}\, 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{\operatorname{arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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