Optimal. Leaf size=39 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^2}{x}-2 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+2 b^2 x \]
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Rubi [A] time = 0.0243156, antiderivative size = 39, 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 = {2168, 2158, 29} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^2}{x}-2 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+2 b^2 x \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^2} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{x}+(2 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=2 b^2 x-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{x}-\left (2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=2 b^2 x-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{x}-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0449575, size = 37, normalized size = 0.95 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^2}{x}+2 b (\log (x)+1) \tanh ^{-1}(\tanh (a+b x))-2 b^2 x \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 42, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{x}}-2\,\ln \left ( x \right ) x{b}^{2}+2\,\ln \left ( x \right ){\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) b+2\,{b}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19392, size = 73, normalized size = 1.87 \begin{align*} 2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) \log \left (x\right ) - 2 \,{\left (b{\left (x + \frac{a}{b}\right )} \log \left (x\right ) - b{\left (x + \frac{a \log \left (x\right )}{b}\right )}\right )} b - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48802, size = 49, normalized size = 1.26 \begin{align*} \frac{b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}}{x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13201, size = 28, normalized size = 0.72 \begin{align*} b^{2} x + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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