Optimal. Leaf size=19 \[ \frac{1}{2} \log \left (1-x^2\right )+x \tanh ^{-1}\left (\frac{1}{x}\right ) \]
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Rubi [A] time = 0.006243, 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 = {6091, 263, 260} \[ \frac{1}{2} \log \left (1-x^2\right )+x \tanh ^{-1}\left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6091
Rule 263
Rule 260
Rubi steps
\begin{align*} \int \tanh ^{-1}\left (\frac{1}{x}\right ) \, dx &=x \tanh ^{-1}\left (\frac{1}{x}\right )+\int \frac{1}{\left (1-\frac{1}{x^2}\right ) x} \, dx\\ &=x \tanh ^{-1}\left (\frac{1}{x}\right )+\int \frac{x}{-1+x^2} \, dx\\ &=x \tanh ^{-1}\left (\frac{1}{x}\right )+\frac{1}{2} \log \left (1-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0020759, size = 17, normalized size = 0.89 \[ \frac{1}{2} \log \left (x^2-1\right )+x \tanh ^{-1}\left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 30, normalized size = 1.6 \begin{align*} x{\it Artanh} \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 = 0.954859, size = 20, normalized size = 1.05 \begin{align*} x \operatorname{artanh}\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 = 1.60687, size = 63, normalized size = 3.32 \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 = 0.258037, size = 15, normalized size = 0.79 \begin{align*} x \operatorname{atanh}{\left (\frac{1}{x} \right )} + \log{\left (x + 1 \right )} - \operatorname{atanh}{\left (\frac{1}{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17986, size = 38, normalized size = 2. \begin{align*} \frac{1}{2} \, x \log \left (-\frac{\frac{1}{x} + 1}{\frac{1}{x} - 1}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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