Optimal. Leaf size=31 \[ \frac{\coth ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0137358, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2167} \[ \frac{\coth ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^5} \, dx &=\frac{\coth ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0222898, size = 50, normalized size = 1.61 \[ -\frac{b^2 x^2 \coth ^{-1}(\tanh (a+b x))+b x \coth ^{-1}(\tanh (a+b x))^2+\coth ^{-1}(\tanh (a+b x))^3+b^3 x^3}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.347, size = 17235, normalized size = 556. \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 = 1.59718, size = 72, normalized size = 2.32 \begin{align*} -\frac{1}{4} \, b{\left (\frac{b^{2}}{x} + \frac{b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\right )} - \frac{b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{3}} - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60225, size = 112, normalized size = 3.61 \begin{align*} -\frac{16 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} - 3 \, \pi ^{2} a + 4 \, a^{3} - 4 \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x}{16 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.14484, size = 56, normalized size = 1.81 \begin{align*} - \frac{b^{3}}{4 x} - \frac{b^{2} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{2}} - \frac{b \operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{3}} - \frac{\operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{4}} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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