Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \]
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Rubi [A] time = 0.0439774, 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} \[ b x-\log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \]
Antiderivative was successfully verified.
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Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))}{x} \, dx &=b x-\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac{1}{x} \, dx\\ &=b x-\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0161214, size = 19, normalized size = 0.9 \[ \log (x) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )+b x \]
Antiderivative was successfully verified.
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Maple [C] time = 0.189, size = 354, normalized size = 16.9 \begin{align*} \ln \left ( x \right ) \ln \left ({{\rm e}^{bx+a}} \right ) -\ln \left ( x \right ) xb+bx+{\frac{i}{2}}\pi \,\ln \left ( x \right ){\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \left ({\it csgn} \left ( i{{\rm e}^{2\,bx+2\,a}} \right ) \right ) ^{2}-{\frac{i}{2}}\pi \,\ln \left ( x \right ) +{\frac{i}{4}}\pi \,\ln \left ( x \right ){\it csgn} \left ({\frac{i}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) \left ({\it csgn} \left ({\frac{i{{\rm e}^{2\,bx+2\,a}}}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) \right ) ^{2}+{\frac{i}{4}}\pi \,\ln \left ( x \right ){\it csgn} \left ( i{{\rm e}^{2\,bx+2\,a}} \right ) \left ({\it csgn} \left ({\frac{i{{\rm e}^{2\,bx+2\,a}}}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) \right ) ^{2}-{\frac{i}{4}}\pi \,\ln \left ( x \right ) \left ({\it csgn} \left ( i{{\rm e}^{2\,bx+2\,a}} \right ) \right ) ^{3}-{\frac{i}{4}}\pi \,\ln \left ( x \right ) \left ({\it csgn} \left ({\frac{i{{\rm e}^{2\,bx+2\,a}}}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) \right ) ^{3}+{\frac{i}{2}}\pi \,\ln \left ( x \right ) \left ({\it csgn} \left ({\frac{i}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) \right ) ^{2}-{\frac{i}{2}}\pi \,\ln \left ( x \right ) \left ({\it csgn} \left ({\frac{i}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) \right ) ^{3}-{\frac{i}{4}}\pi \,\ln \left ( x \right ){\it csgn} \left ({\frac{i}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ){\it csgn} \left ( i{{\rm e}^{2\,bx+2\,a}} \right ){\it csgn} \left ({\frac{i{{\rm e}^{2\,bx+2\,a}}}{{{\rm e}^{2\,bx+2\,a}}+1}} \right ) -{\frac{i}{4}}\pi \,\ln \left ( x \right ) \left ({\it csgn} \left ( i{{\rm e}^{bx+a}} \right ) \right ) ^{2}{\it csgn} \left ( i{{\rm e}^{2\,bx+2\,a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977923, 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{arcoth}\left (\tanh \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 = 1.62091, 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{acoth}{\left (\tanh{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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