Integrand size = 13, antiderivative size = 23 \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=x \cot ^{-1}(x)+\frac {1}{2} \cot ^{-1}(x)^2+\frac {1}{2} \log \left (1+x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5037, 4931, 266, 5005} \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
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Rule 266
Rule 4931
Rule 5005
Rule 5037
Rubi steps \begin{align*} \text {integral}& = \int \cot ^{-1}(x) \, dx-\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx \\ & = x \cot ^{-1}(x)+\frac {1}{2} \cot ^{-1}(x)^2+\int \frac {x}{1+x^2} \, dx \\ & = x \cot ^{-1}(x)+\frac {1}{2} \cot ^{-1}(x)^2+\frac {1}{2} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=x \cot ^{-1}(x)+\frac {1}{2} \cot ^{-1}(x)^2+\frac {1}{2} \log \left (1+x^2\right ) \]
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Time = 0.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(x \,\operatorname {arccot}\left (x \right )+\frac {\operatorname {arccot}\left (x \right )^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(20\) |
default | \(-\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+x \,\operatorname {arccot}\left (x \right )+\frac {\ln \left (x^{2}+1\right )}{2}-\frac {\arctan \left (x \right )^{2}}{2}\) | \(26\) |
parts | \(-\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+x \,\operatorname {arccot}\left (x \right )+\frac {\ln \left (x^{2}+1\right )}{2}-\frac {\arctan \left (x \right )^{2}}{2}\) | \(26\) |
risch | \(-\frac {\ln \left (i x +1\right )^{2}}{8}+\left (\frac {i x}{2}+\frac {\ln \left (-i x +1\right )}{4}\right ) \ln \left (i x +1\right )-\frac {\ln \left (-i x +1\right )^{2}}{8}-\frac {i \ln \left (-i x +1\right ) x}{2}+\frac {\pi x}{2}+\frac {\ln \left (x^{2}+1\right )}{2}-\frac {\pi \arctan \left (x \right )}{2}\) | \(74\) |
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=x \operatorname {arccot}\left (x\right ) + \frac {1}{2} \, \operatorname {arccot}\left (x\right )^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=x \operatorname {acot}{\left (x \right )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {\operatorname {acot}^{2}{\left (x \right )}}{2} \]
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none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx={\left (x - \arctan \left (x\right )\right )} \operatorname {arccot}\left (x\right ) - \frac {1}{2} \, \arctan \left (x\right )^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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\[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{2} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]
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Time = 0.70 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {{\mathrm {acot}\left (x\right )}^2}{2}+x\,\mathrm {acot}\left (x\right )+\frac {\ln \left (x^2+1\right )}{2} \]
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