Optimal. Leaf size=55 \[ -\frac{b^2 \coth ^{-1}(\tanh (a+b x))}{x}-\frac{b \coth ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \log (x) \]
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Rubi [A] time = 0.0381152, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 29} \[ -\frac{b^2 \coth ^{-1}(\tanh (a+b x))}{x}-\frac{b \coth ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 2168
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^4} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}+b \int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^3} \, dx\\ &=-\frac{b \coth ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^2 \int \frac{\coth ^{-1}(\tanh (a+b x))}{x^2} \, dx\\ &=-\frac{b^2 \coth ^{-1}(\tanh (a+b x))}{x}-\frac{b \coth ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \int \frac{1}{x} \, dx\\ &=-\frac{b^2 \coth ^{-1}(\tanh (a+b x))}{x}-\frac{b \coth ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0243607, size = 60, normalized size = 1.09 \[ \frac{-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))-3 b x \coth ^{-1}(\tanh (a+b x))^2-2 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3 (6 \log (x)+11)}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.385, size = 17237, normalized size = 313.4 \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 = 1.59677, size = 70, normalized size = 1.27 \begin{align*}{\left (b^{2} \log \left (x\right ) - \frac{b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{x}\right )} b - \frac{b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x^{2}} - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65861, size = 120, normalized size = 2.18 \begin{align*} \frac{24 \, b^{3} x^{3} \log \left (x\right ) - 72 \, a b^{2} x^{2} + 6 \, \pi ^{2} a - 8 \, a^{3} + 9 \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x}{24 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21213, size = 51, normalized size = 0.93 \begin{align*} b^{3} \log{\left (x \right )} - \frac{b^{2} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{x} - \frac{b \operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{2 x^{2}} - \frac{\operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{3}} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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