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 = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 260
Rule 263
Rule 6091
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.64, size = 22, normalized size = 1.16 \[ \frac {1}{2} \, x \log \left (\frac {x + 1}{x - 1}\right ) + \frac {1}{2} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 101, normalized size = 5.32 \[ \frac {\log \left (-\frac {\frac {\frac {x + 1}{x - 1} - 1}{\frac {x + 1}{x - 1} + 1} + 1}{\frac {\frac {x + 1}{x - 1} - 1}{\frac {x + 1}{x - 1} + 1} - 1}\right )}{\frac {x + 1}{x - 1} - 1} + \log \left (\frac {{\left | x + 1 \right |}}{{\left | x - 1 \right |}}\right ) - \log \left ({\left | \frac {x + 1}{x - 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 1.58 \[ x \arctanh \left (\frac {1}{x}\right )-\ln \left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}-1\right )}{2}+\frac {\ln \left (\frac {1}{x}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 15, normalized size = 0.79 \[ x \operatorname {artanh}\left (\frac {1}{x}\right ) + \frac {1}{2} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 15, normalized size = 0.79 \[ \frac {\ln \left (x^2-1\right )}{2}+x\,\mathrm {atanh}\left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 15, normalized size = 0.79 \[ x \operatorname {atanh}{\left (\frac {1}{x} \right )} + \log {\left (x + 1 \right )} - \operatorname {atanh}{\left (\frac {1}{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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