Optimal. Leaf size=17 \[ b \log (x)-\frac{\tanh ^{-1}(\coth (a+b x))}{x} \]
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Rubi [A] time = 0.0082163, antiderivative size = 17, 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 = {2168, 29} \[ b \log (x)-\frac{\tanh ^{-1}(\coth (a+b x))}{x} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\coth (a+b x))}{x^2} \, dx &=-\frac{\tanh ^{-1}(\coth (a+b x))}{x}+b \int \frac{1}{x} \, dx\\ &=-\frac{\tanh ^{-1}(\coth (a+b x))}{x}+b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0154534, size = 18, normalized size = 1.06 \[ -\frac{\tanh ^{-1}(\coth (a+b x))}{x}+b \log (x)+b \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 18, normalized size = 1.1 \begin{align*} -{\frac{{\it Artanh} \left ({\rm coth} \left (bx+a\right ) \right ) }{x}}+b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13869, size = 23, normalized size = 1.35 \begin{align*} b \log \left (x\right ) - \frac{\operatorname{artanh}\left (\coth \left (b x + a\right )\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08442, size = 27, normalized size = 1.59 \begin{align*} \frac{b x \log \left (x\right ) - a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.0947, size = 42, normalized size = 2.47 \begin{align*} \begin{cases} \frac{\left \langle - \frac{\pi }{2}, \frac{\pi }{2}\right \rangle i}{x} & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\b \log{\left (x \right )} - \frac{\operatorname{atanh}{\left (\frac{1}{\tanh{\left (a + b x \right )}} \right )}}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18725, size = 95, normalized size = 5.59 \begin{align*} b \log \left ({\left | x \right |}\right ) - \frac{\log \left (-\frac{\frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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