Optimal. Leaf size=68 \[ -3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+\frac{3}{2} b \coth ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \]
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Rubi [A] time = 0.0435156, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2168, 2159, 2158, 29} \[ -3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+\frac{3}{2} b \coth ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^2} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+(3 b) \int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=\frac{3}{2} b \coth ^{-1}(\tanh (a+b x))^2-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}-\left (3 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\coth ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac{3}{2} b \coth ^{-1}(\tanh (a+b x))^2-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+\left (3 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac{1}{x} \, dx\\ &=-3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac{3}{2} b \coth ^{-1}(\tanh (a+b x))^2-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+3 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.039466, size = 62, normalized size = 0.91 \[ -6 b^2 x \log (x) \coth ^{-1}(\tanh (a+b x))-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+3 b (\log (x)+1) \coth ^{-1}(\tanh (a+b x))^2+\frac{3}{2} b^3 x^2 (2 \log (x)-1) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.415, size = 7683, normalized size = 113. \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.08232, size = 166, normalized size = 2.44 \begin{align*} 3 \, b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right ) - \frac{3}{2} \,{\left (2 \, \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right ) -{\left (b x^{2} +{\left (2 i \, \pi + 4 \, a\right )} x + 2 \,{\left (-\frac{i \, \pi{\left (b x + a\right )}}{b} - \frac{{\left (b x + a\right )}^{2}}{b}\right )} \log \left (x\right ) + \frac{2 \, \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right )}{b} + \frac{2 \,{\left (i \, \pi a + a^{2}\right )} \log \left (x\right )}{b}\right )} b\right )} b - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72542, size = 115, normalized size = 1.69 \begin{align*} \frac{2 \, b^{3} x^{3} + 12 \, a b^{2} x^{2} + 3 \, \pi ^{2} a - 4 \, a^{3} - 3 \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x \log \left (x\right )}{4 \, 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{acoth}^{3}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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