Integrand size = 12, antiderivative size = 56 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {\arctan (x)}{4} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5013, 5051, 205, 209} \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {\arctan (x)}{4}-\frac {x}{4 \left (x^2+1\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{6} \cot ^{-1}(x)^3 \]
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Rule 205
Rule 209
Rule 5013
Rule 5051
Rubi steps \begin{align*} \text {integral}& = \frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3+\int \frac {x \cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx \\ & = -\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {1}{2} \int \frac {1}{\left (1+x^2\right )^2} \, dx \\ & = -\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {1}{4} \int \frac {1}{1+x^2} \, dx \\ & = -\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {\arctan (x)}{4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {6 \cot ^{-1}(x)-6 x \cot ^{-1}(x)^2+2 \left (1+x^2\right ) \cot ^{-1}(x)^3+3 \left (x+\left (1+x^2\right ) \arctan (x)\right )}{12 \left (1+x^2\right )} \]
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Time = 0.96 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\operatorname {arccot}\left (x \right )^{2} x}{2 x^{2}+2}+\frac {\operatorname {arccot}\left (x \right )^{2} \arctan \left (x \right )}{2}-\frac {\pi \operatorname {arccot}\left (x \right )^{2}}{4}+\frac {\operatorname {arccot}\left (x \right )^{3}}{3}+\frac {x^{2} \operatorname {arccot}\left (x \right )}{2 x^{2}+2}-\frac {x}{4 \left (x^{2}+1\right )}-\frac {\operatorname {arccot}\left (x \right )}{4}\) | \(65\) |
parts | \(\frac {\operatorname {arccot}\left (x \right )^{2} x}{2 x^{2}+2}+\frac {\operatorname {arccot}\left (x \right )^{2} \arctan \left (x \right )}{2}-\frac {\pi \operatorname {arccot}\left (x \right )^{2}}{4}+\frac {\operatorname {arccot}\left (x \right )^{3}}{3}+\frac {x^{2} \operatorname {arccot}\left (x \right )}{2 x^{2}+2}-\frac {x}{4 \left (x^{2}+1\right )}-\frac {\operatorname {arccot}\left (x \right )}{4}\) | \(65\) |
risch | \(\frac {i \ln \left (i x +1\right )^{3}}{48}+\frac {\left (-i x^{2} \ln \left (-i x +1\right )+\pi \,x^{2}-i \ln \left (-i x +1\right )+\pi -2 x \right ) \ln \left (i x +1\right )^{2}}{16 x^{2}+16}-\frac {\left (-i x^{2} \ln \left (-i x +1\right )^{2}-i \ln \left (-i x +1\right )^{2}-4 \ln \left (-i x +1\right ) x +2 \ln \left (-i x +1\right ) \pi \,x^{2}+2 \pi \ln \left (-i x +1\right )-4 i \pi x +4 i\right ) \ln \left (i x +1\right )}{16 \left (i+x \right ) \left (x -i\right )}-\frac {i \left (-3 \ln \left (i+x \right ) \pi ^{2} x^{2}+3 \ln \left (x -i\right ) \pi ^{2} x^{2}+x^{2} \ln \left (-i x +1\right )^{3}+3 i \pi \ln \left (-i x +1\right )^{2}-3 \ln \left (i+x \right ) \pi ^{2}+6 \ln \left (i+x \right ) x^{2}+3 \ln \left (x -i\right ) \pi ^{2}-6 \ln \left (x -i\right ) x^{2}+12 \pi \ln \left (-i x +1\right ) x +\ln \left (-i x +1\right )^{3}-12 i \pi -12 i x +3 i \pi \,x^{2} \ln \left (-i x +1\right )^{2}+6 \ln \left (i+x \right )-6 \ln \left (x -i\right )-12 \ln \left (-i x +1\right )-6 i \ln \left (-i x +1\right )^{2} x +6 i \pi ^{2} x \right )}{48 \left (i+x \right ) \left (x -i\right )}\) | \(349\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {2 \, {\left (x^{2} + 1\right )} \operatorname {arccot}\left (x\right )^{3} - 6 \, x \operatorname {arccot}\left (x\right )^{2} - 3 \, {\left (x^{2} - 1\right )} \operatorname {arccot}\left (x\right ) + 3 \, x}{12 \, {\left (x^{2} + 1\right )}} \]
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\[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\int \frac {\operatorname {acot}^{2}{\left (x \right )}}{\left (x^{2} + 1\right )^{2}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} + \arctan \left (x\right )\right )} \operatorname {arccot}\left (x\right )^{2} + \frac {{\left ({\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 1\right )} \operatorname {arccot}\left (x\right )}{2 \, {\left (x^{2} + 1\right )}} + \frac {2 \, {\left (x^{2} + 1\right )} \arctan \left (x\right )^{3} - 3 \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) - 3 \, x}{12 \, {\left (x^{2} + 1\right )}} \]
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\[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )^{2}}{{\left (x^{2} + 1\right )}^{2}} \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\frac {x\,{\mathrm {acot}\left (x\right )}^2}{2\,\left (x^2+1\right )}-\frac {{\mathrm {acot}\left (x\right )}^3}{6}-\frac {x}{4\,\left (x^2+1\right )}-\frac {\mathrm {acot}\left (x\right )}{2\,\left (x^2+1\right )}-\frac {\mathrm {atan}\left (x\right )}{4} \]
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