Optimal. Leaf size=60 \[ -3 b^2 \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\frac{3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}+3 b^3 x \]
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Rubi [A] time = 0.0411344, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2168, 2158, 29} \[ -3 b^2 \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\frac{3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}+3 b^3 x \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^3} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}+\frac{1}{2} (3 b) \int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^2} \, dx\\ &=-\frac{3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}+\left (3 b^2\right ) \int \frac{\coth ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=3 b^3 x-\frac{3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\left (3 b^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=3 b^3 x-\frac{3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-3 b^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0363074, size = 66, normalized size = 1.1 \[ 3 b^2 \log (x) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )-\frac{\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3}{2 x^2}-\frac{3 b \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2}{x}+b^3 x \]
Antiderivative was successfully verified.
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Maple [C] time = 0.426, size = 7366, normalized size = 122.8 \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 [A] time = 1.40292, size = 97, normalized size = 1.62 \begin{align*} 3 \,{\left (b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right ) \log \left (x\right ) -{\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\right )} b - \frac{3 \, b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x} - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61919, size = 117, normalized size = 1.95 \begin{align*} \frac{8 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} \log \left (x\right ) + 3 \, \pi ^{2} a - 4 \, a^{3} + 6 \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x}{8 \, x^{2}} \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^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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