Optimal. Leaf size=34 \[ -\frac{1}{4 \left (x^2+1\right )}+\frac{x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{4} \cot ^{-1}(x)^2 \]
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Rubi [A] time = 0.0145949, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4893, 261} \[ -\frac{1}{4 \left (x^2+1\right )}+\frac{x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{4} \cot ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 4893
Rule 261
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=\frac{x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{4} \cot ^{-1}(x)^2+\frac{1}{2} \int \frac{x}{\left (1+x^2\right )^2} \, dx\\ &=-\frac{1}{4 \left (1+x^2\right )}+\frac{x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{4} \cot ^{-1}(x)^2\\ \end{align*}
Mathematica [A] time = 0.0127626, size = 28, normalized size = 0.82 \[ -\frac{\left (x^2+1\right ) \cot ^{-1}(x)^2-2 x \cot ^{-1}(x)+1}{4 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 35, normalized size = 1. \begin{align*}{\frac{x{\rm arccot} \left (x\right )}{2\,{x}^{2}+2}}+{\frac{{\rm arccot} \left (x\right )\arctan \left ( x \right ) }{2}}-{\frac{1}{4\,{x}^{2}+4}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46364, size = 51, normalized size = 1.5 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{x^{2} + 1} + \arctan \left (x\right )\right )} \operatorname{arccot}\left (x\right ) + \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 1}{4 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1514, size = 81, normalized size = 2.38 \begin{align*} -\frac{{\left (x^{2} + 1\right )} \operatorname{arccot}\left (x\right )^{2} - 2 \, x \operatorname{arccot}\left (x\right ) + 1}{4 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (x \right )}}{\left (x^{2} + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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