Optimal. Leaf size=30 \[ \frac{1}{2} \log \left (x^2+1\right )-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x} \]
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Rubi [A] time = 0.0557802, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {4919, 4853, 266, 36, 29, 31, 4885} \[ \frac{1}{2} \log \left (x^2+1\right )-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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Rule 4919
Rule 4853
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4885
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(x)}{x^2 \left (1+x^2\right )} \, dx &=\int \frac{\cot ^{-1}(x)}{x^2} \, dx-\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac{\cot ^{-1}(x)}{x}+\frac{1}{2} \cot ^{-1}(x)^2-\int \frac{1}{x \left (1+x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(x)}{x}+\frac{1}{2} \cot ^{-1}(x)^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(x)}{x}+\frac{1}{2} \cot ^{-1}(x)^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(x)}{x}+\frac{1}{2} \cot ^{-1}(x)^2-\log (x)+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0158508, size = 30, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+1\right )-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 33, normalized size = 1.1 \begin{align*} -{\rm arccot} \left (x\right )\arctan \left ( x \right ) -{\frac{{\rm arccot} \left (x\right )}{x}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-\ln \left ( x \right ) -{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50688, size = 39, normalized size = 1.3 \begin{align*} -{\left (\frac{1}{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 ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94416, size = 90, normalized size = 3. \begin{align*} \frac{x \operatorname{arccot}\left (x\right )^{2} + x \log \left (x^{2} + 1\right ) - 2 \, x \log \left (x\right ) - 2 \, \operatorname{arccot}\left (x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.654051, size = 22, normalized size = 0.73 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{\operatorname{acot}^{2}{\left (x \right )}}{2} - \frac{\operatorname{acot}{\left (x \right )}}{x} \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 )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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