Optimal. Leaf size=64 \[ \frac{\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac{b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0335485, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac{\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac{b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^6} \, dx &=\frac{\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac{b \int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^5} \, dx}{5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac{\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0339501, size = 54, normalized size = 0.84 \[ -\frac{2 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+3 b x \coth ^{-1}(\tanh (a+b x))^2+4 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3}{20 x^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.273, size = 17234, normalized size = 269.3 \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 = 1.59881, size = 73, normalized size = 1.14 \begin{align*} -\frac{1}{20} \, b{\left (\frac{b^{2}}{x^{2}} + \frac{2 \, b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{x^{3}}\right )} - \frac{3 \, b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{20 \, x^{4}} - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69803, size = 116, normalized size = 1.81 \begin{align*} -\frac{40 \, b^{3} x^{3} + 80 \, a b^{2} x^{2} - 12 \, \pi ^{2} a + 16 \, a^{3} - 15 \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x}{80 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.61206, size = 60, normalized size = 0.94 \begin{align*} - \frac{b^{3}}{20 x^{2}} - \frac{b^{2} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{10 x^{3}} - \frac{3 b \operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{20 x^{4}} - \frac{\operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{5 x^{5}} \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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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