Optimal. Leaf size=15 \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]
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Rubi [A] time = 0.0033438, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4847, 260} \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4847
Rule 260
Rubi steps
\begin{align*} \int \cot ^{-1}(x) \, dx &=x \cot ^{-1}(x)+\int \frac{x}{1+x^2} \, dx\\ &=x \cot ^{-1}(x)+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0014764, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 14, normalized size = 0.9 \begin{align*} x{\rm arccot} \left (x\right )+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931492, size = 18, normalized size = 1.2 \begin{align*} x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.513946, size = 43, normalized size = 2.87 \begin{align*} x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.18563, size = 12, normalized size = 0.8 \begin{align*} x \operatorname{acot}{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08174, size = 20, normalized size = 1.33 \begin{align*} x \arctan \left (\frac{1}{x}\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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