Integrand size = 13, antiderivative size = 40 \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {x^2}{6}-x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2-\frac {2}{3} \log \left (1+x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5037, 4947, 272, 45, 4931, 266, 5005} \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{3} x^3 \cot ^{-1}(x)+\frac {x^2}{6}-\frac {2}{3} \log \left (x^2+1\right )-x \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2 \]
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Rule 45
Rule 266
Rule 272
Rule 4931
Rule 4947
Rule 5005
Rule 5037
Rubi steps \begin{align*} \text {integral}& = \int x^2 \cot ^{-1}(x) \, dx-\int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx \\ & = \frac {1}{3} x^3 \cot ^{-1}(x)+\frac {1}{3} \int \frac {x^3}{1+x^2} \, dx-\int \cot ^{-1}(x) \, dx+\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx \\ & = -x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2+\frac {1}{6} \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,x^2\right )-\int \frac {x}{1+x^2} \, dx \\ & = -x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2-\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{6} \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^2}{6}-x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2-\frac {2}{3} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80 \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{6} \left (x^2+2 x \left (-3+x^2\right ) \cot ^{-1}(x)-3 \cot ^{-1}(x)^2-4 \log \left (1+x^2\right )\right ) \]
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Time = 0.48 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {x^{3} \operatorname {arccot}\left (x \right )}{3}+\frac {x^{2}}{6}-x \,\operatorname {arccot}\left (x \right )-\frac {\operatorname {arccot}\left (x \right )^{2}}{2}-\frac {2 \ln \left (x^{2}+1\right )}{3}-\frac {1}{3}\) | \(34\) |
default | \(\frac {x^{3} \operatorname {arccot}\left (x \right )}{3}-x \,\operatorname {arccot}\left (x \right )+\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+\frac {x^{2}}{6}-\frac {2 \ln \left (x^{2}+1\right )}{3}+\frac {\arctan \left (x \right )^{2}}{2}\) | \(38\) |
parts | \(\frac {x^{3} \operatorname {arccot}\left (x \right )}{3}-x \,\operatorname {arccot}\left (x \right )+\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+\frac {x^{2}}{6}-\frac {2 \ln \left (x^{2}+1\right )}{3}+\frac {\arctan \left (x \right )^{2}}{2}\) | \(38\) |
risch | \(\frac {\ln \left (i x +1\right )^{2}}{8}+\left (\frac {i x^{3}}{6}-\frac {i x}{2}-\frac {\ln \left (-i x +1\right )}{4}\right ) \ln \left (i x +1\right )+\frac {\ln \left (-i x +1\right )^{2}}{8}-\frac {i x^{3} \ln \left (-i x +1\right )}{6}+\frac {i \ln \left (-i x +1\right ) x}{2}+\frac {\pi \,x^{3}}{6}-\frac {\pi x}{2}+\frac {x^{2}}{6}-\frac {2 \ln \left (x^{2}+1\right )}{3}+\frac {\pi \arctan \left (x \right )}{2}\) | \(104\) |
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{6} \, x^{2} + \frac {1}{3} \, {\left (x^{3} - 3 \, x\right )} \operatorname {arccot}\left (x\right ) - \frac {1}{2} \, \operatorname {arccot}\left (x\right )^{2} - \frac {2}{3} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {x^{3} \operatorname {acot}{\left (x \right )}}{3} + \frac {x^{2}}{6} - x \operatorname {acot}{\left (x \right )} - \frac {2 \log {\left (x^{2} + 1 \right )}}{3} - \frac {\operatorname {acot}^{2}{\left (x \right )}}{2} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{6} \, x^{2} + \frac {1}{3} \, {\left (x^{3} - 3 \, x + 3 \, \arctan \left (x\right )\right )} \operatorname {arccot}\left (x\right ) + \frac {1}{2} \, \arctan \left (x\right )^{2} - \frac {2}{3} \, \log \left (x^{2} + 1\right ) \]
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\[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{4} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]
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Time = 0.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80 \[ \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {x^3\,\mathrm {acot}\left (x\right )}{3}-\frac {2\,\ln \left (x^2+1\right )}{3}-\frac {{\mathrm {acot}\left (x\right )}^2}{2}-x\,\mathrm {acot}\left (x\right )+\frac {x^2}{6} \]
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