Optimal. Leaf size=19 \[ \frac{1}{2} \log \left (1-x^2\right )+x \coth ^{-1}\left (\frac{1}{x}\right ) \]
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Rubi [A] time = 0.0060927, antiderivative size = 19, 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 = {6092, 263, 260} \[ \frac{1}{2} \log \left (1-x^2\right )+x \coth ^{-1}\left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6092
Rule 263
Rule 260
Rubi steps
\begin{align*} \int \coth ^{-1}\left (\frac{1}{x}\right ) \, dx &=x \coth ^{-1}\left (\frac{1}{x}\right )+\int \frac{1}{\left (1-\frac{1}{x^2}\right ) x} \, dx\\ &=x \coth ^{-1}\left (\frac{1}{x}\right )+\int \frac{x}{-1+x^2} \, dx\\ &=x \coth ^{-1}\left (\frac{1}{x}\right )+\frac{1}{2} \log \left (1-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0018144, size = 19, normalized size = 1. \[ \frac{1}{2} \log \left (1-x^2\right )+x \coth ^{-1}\left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 30, normalized size = 1.6 \begin{align*} x{\rm arccoth} \left ({x}^{-1}\right )+{\frac{\ln \left ({x}^{-1}-1 \right ) }{2}}-\ln \left ({x}^{-1} \right ) +{\frac{\ln \left ({x}^{-1}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976526, size = 20, normalized size = 1.05 \begin{align*} x \operatorname{arcoth}\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 = 1.85874, size = 65, normalized size = 3.42 \begin{align*} \frac{1}{2} \, x \log \left (-\frac{x + 1}{x - 1}\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 = 0.254185, size = 15, normalized size = 0.79 \begin{align*} x \operatorname{acoth}{\left (\frac{1}{x} \right )} + \log{\left (x + 1 \right )} - \operatorname{acoth}{\left (\frac{1}{x} \right )} \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 \operatorname{arcoth}\left (\frac{1}{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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