Optimal. Leaf size=36 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b \tanh ^{-1}(\tanh (a+b x))}{x}+b^2 \log (x) \]
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Rubi [A] time = 0.0211679, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 29} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b \tanh ^{-1}(\tanh (a+b x))}{x}+b^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 2168
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^3} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^2} \, dx\\ &=-\frac{b \tanh ^{-1}(\tanh (a+b x))}{x}-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^2 \int \frac{1}{x} \, dx\\ &=-\frac{b \tanh ^{-1}(\tanh (a+b x))}{x}-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0326993, size = 42, normalized size = 1.17 \[ -\frac{2 b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2-b^2 x^2 (2 \log (x)+3)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 35, normalized size = 1. \begin{align*} -{\frac{b{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{x}}-{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{2\,{x}^{2}}}+{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37652, size = 46, normalized size = 1.28 \begin{align*} b^{2} \log \left (x\right ) - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5044, size = 59, normalized size = 1.64 \begin{align*} \frac{2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.822608, size = 32, normalized size = 0.89 \begin{align*} b^{2} \log{\left (x \right )} - \frac{b \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{x} - \frac{\operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18836, size = 30, normalized size = 0.83 \begin{align*} b^{2} \log \left ({\left | x \right |}\right ) - \frac{4 \, a b x + a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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