Optimal. Leaf size=49 \[ \frac {1}{2} i \text {Li}_2\left (\frac {2}{1-i x}-1\right )+\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x) \]
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Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 2447
Rule 4869
Rule 4925
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*}
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Mathematica [B] time = 0.06, size = 251, normalized size = 5.12 \[ -\frac {1}{4} i \text {Li}_2\left (-\frac {1}{2} i (i-x)\right )-\frac {1}{2} i \text {Li}_2\left (-\frac {i}{x}\right )+\frac {1}{2} i \text {Li}_2\left (\frac {i}{x}\right )+\frac {1}{4} i \text {Li}_2\left (-\frac {1}{2} i (x+i)\right )+\frac {1}{8} i \log ^2(-x+i)-\frac {1}{8} i \log ^2(x+i)-\frac {1}{4} i \log \left (-\frac {-x+i}{x}\right ) \log (-x+i)-\frac {1}{4} i \log \left (-\frac {1}{2} i (x+i)\right ) \log (-x+i)+\frac {1}{4} i \log \left (\frac {x+i}{x}\right ) \log (-x+i)+\frac {1}{4} i \log \left (-\frac {1}{2} i (-x+i)\right ) \log (x+i)-\frac {1}{4} i \log \left (-\frac {-x+i}{x}\right ) \log (x+i)+\frac {1}{4} i \log (x+i) \log \left (\frac {x+i}{x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\relax (x)}{x^{3} + x}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\relax (x)}{{\left (x^{2} + 1\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 163, normalized size = 3.33 \[ \mathrm {arccot}\relax (x ) \ln \relax (x )-\frac {\mathrm {arccot}\relax (x ) \ln \left (x^{2}+1\right )}{2}-\frac {i \ln \relax (x ) \ln \left (i x +1\right )}{2}+\frac {i \ln \relax (x ) \ln \left (-i x +1\right )}{2}-\frac {i \dilog \left (i x +1\right )}{2}+\frac {i \dilog \left (-i x +1\right )}{2}+\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (x -i\right )^{2}}{8}-\frac {i \dilog \left (-\frac {i \left (x +i\right )}{2}\right )}{4}-\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )}{4}-\frac {i \ln \left (x +i\right ) \ln \left (x^{2}+1\right )}{4}+\frac {i \ln \left (x +i\right )^{2}}{8}+\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{4}+\frac {i \ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{4} \]
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 \[ \int \frac {\operatorname {arccot}\relax (x)}{{\left (x^{2} + 1\right )} x}\,{d x} \]
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 \[ \int \frac {\mathrm {acot}\relax (x)}{x\,\left (x^2+1\right )} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\relax (x )}}{x \left (x^{2} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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