Optimal. Leaf size=49 \[ -b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2+\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
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Rubi [A] time = 0.033532, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2159, 2158, 29} \[ -b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2+\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx &=\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=-b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2+\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0408171, size = 53, normalized size = 1.08 \[ \frac{1}{2} (a+b x)^2-(a+b x) \left (-2 \tanh ^{-1}(\tanh (a+b x))+a+2 b x\right )+\log (b x) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 55, normalized size = 1.1 \begin{align*} \ln \left ( x \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}+{b}^{2}{x}^{2}\ln \left ( x \right ) -{\frac{3\,{b}^{2}{x}^{2}}{2}}-2\,b{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \ln \left ( x \right ) x+2\,b{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.35913, size = 27, normalized size = 0.55 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47833, size = 49, normalized size = 1. \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \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}^{2}{\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 [A] time = 1.12558, size = 28, normalized size = 0.57 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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