Optimal. Leaf size=17 \[ b \log (x)-\frac{\coth ^{-1}(\tanh (a+b x))}{x} \]
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Rubi [A] time = 0.0092769, 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} \[ b \log (x)-\frac{\coth ^{-1}(\tanh (a+b x))}{x} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))}{x^2} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))}{x}+b \int \frac{1}{x} \, dx\\ &=-\frac{\coth ^{-1}(\tanh (a+b x))}{x}+b \log (x)\\ \end{align*}
Mathematica [A] time = 0.016392, size = 18, normalized size = 1.06 \[ -\frac{\coth ^{-1}(\tanh (a+b x))}{x}+b \log (x)+b \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 20, normalized size = 1.2 \begin{align*} -{\frac{{\rm arccoth} \left (\tanh \left ( bx+a \right ) \right )}{x}}+b\ln \left ( bx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17533, size = 23, normalized size = 1.35 \begin{align*} b \log \left (x\right ) - \frac{\operatorname{arcoth}\left (\tanh \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 = 1.72301, 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 = 0.314397, size = 14, normalized size = 0.82 \begin{align*} b \log{\left (x \right )} - \frac{\operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{x} \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 \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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