Integrand size = 13, antiderivative size = 49 \[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\frac {1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5045, 4989, 2497} \[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {2}{1-i x}-1\right )+\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x) \]
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Rule 2497
Rule 4989
Rule 5045
Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i x}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=-\frac {1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (1+e^{2 i \cot ^{-1}(x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(x)}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (41 ) = 82\).
Time = 0.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.39
method | result | size |
risch | \(-\frac {\pi \ln \left (x^{2}+1\right )}{4}+\frac {\pi \ln \left (-i x \right )}{2}+\frac {i \ln \left (-i x +1\right )^{2}}{8}+\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}+\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}-\frac {i \ln \left (i x +1\right )^{2}}{8}-\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}-\frac {i \operatorname {dilog}\left (i x +1\right )}{2}\) | \(117\) |
default | \(\operatorname {arccot}\left (x \right ) \ln \left (x \right )-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\) | \(161\) |
parts | \(\operatorname {arccot}\left (x \right ) \ln \left (x \right )-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\) | \(161\) |
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\[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]
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\[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{x \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]
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\[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{x\,\left (x^2+1\right )} \,d x \]
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