Optimal. Leaf size=17 \[ x \cot ^{-1}\left (\frac{1}{x}\right )-\frac{1}{2} \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.005577, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5028, 263, 260} \[ x \cot ^{-1}\left (\frac{1}{x}\right )-\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 5028
Rule 263
Rule 260
Rubi steps
\begin{align*} \int \cot ^{-1}\left (\frac{1}{x}\right ) \, dx &=x \cot ^{-1}\left (\frac{1}{x}\right )-\int \frac{1}{\left (1+\frac{1}{x^2}\right ) x} \, dx\\ &=x \cot ^{-1}\left (\frac{1}{x}\right )-\int \frac{x}{1+x^2} \, dx\\ &=x \cot ^{-1}\left (\frac{1}{x}\right )-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0015833, size = 17, normalized size = 1. \[ x \cot ^{-1}\left (\frac{1}{x}\right )-\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 20, normalized size = 1.2 \begin{align*} x{\rm arccot} \left ({x}^{-1}\right )-{\frac{\ln \left ({x}^{-2}+1 \right ) }{2}}+\ln \left ({x}^{-1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992027, size = 20, normalized size = 1.18 \begin{align*} x \operatorname{arccot}\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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Fricas [A] time = 2.07521, size = 46, normalized size = 2.71 \begin{align*} x \operatorname{arccot}\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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Sympy [A] time = 0.191395, size = 14, normalized size = 0.82 \begin{align*} x \operatorname{acot}{\left (\frac{1}{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 = 1.09883, size = 18, normalized size = 1.06 \begin{align*} x \arctan \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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