Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\tanh ^{-1}(\coth (a+b x))\right ) \]
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Rubi [A] time = 0.0303547, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2158, 29} \[ b x-\log (x) \left (b x-\tanh ^{-1}(\coth (a+b x))\right ) \]
Antiderivative was successfully verified.
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Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\coth (a+b x))}{x} \, dx &=b x-\left (b x-\tanh ^{-1}(\coth (a+b x))\right ) \int \frac{1}{x} \, dx\\ &=b x-\left (b x-\tanh ^{-1}(\coth (a+b x))\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0161014, size = 19, normalized size = 0.9 \[ \log (x) \left (\tanh ^{-1}(\coth (a+b x))-b x\right )+b x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 21, normalized size = 1. \begin{align*} \ln \left ( x \right ){\it Artanh} \left ({\rm coth} \left (bx+a\right ) \right ) -\ln \left ( x \right ) xb+bx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95269, size = 46, normalized size = 2.19 \begin{align*} -b{\left (x + \frac{a}{b}\right )} \log \left (x\right ) + b{\left (x + \frac{a \log \left (x\right )}{b}\right )} + \operatorname{artanh}\left (\coth \left (b x + a\right )\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13451, size = 22, normalized size = 1.05 \begin{align*} b x + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (\coth{\left (a + b x \right )} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16903, size = 20, normalized size = 0.95 \begin{align*} b x + \frac{1}{2} \,{\left (i \, \pi + 2 \, a\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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