Optimal. Leaf size=34 \[ \frac{x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \]
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Rubi [A] time = 0.0149233, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2168, 2157, 30} \[ \frac{x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\coth ^{-1}(\tanh (a+b x))^5}{20 b^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2157
Rule 30
Rubi steps
\begin{align*} \int x \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\int \coth ^{-1}(\tanh (a+b x))^4 \, dx}{4 b}\\ &=\frac{x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{4 b^2}\\ &=\frac{x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\coth ^{-1}(\tanh (a+b x))^5}{20 b^2}\\ \end{align*}
Mathematica [B] time = 0.0736339, size = 99, normalized size = 2.91 \[ \frac{(a+b x) \left (10 \left (2 a^2+a b x-b^2 x^2\right ) \coth ^{-1}(\tanh (a+b x))^2+(4 a-b x) (a+b x)^3-5 (3 a-b x) (a+b x)^2 \coth ^{-1}(\tanh (a+b x))-10 (a-b x) \coth ^{-1}(\tanh (a+b x))^3\right )}{20 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.083, size = 18111, normalized size = 532.7 \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 = 1.55891, size = 73, normalized size = 2.15 \begin{align*} -\frac{1}{2} \, b x^{3} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{1}{2} \, x^{2} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{20} \,{\left (b^{2} x^{5} - 5 \, b x^{4} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50232, size = 117, normalized size = 3.44 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} - \frac{1}{4} \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{3} - \frac{1}{8} \,{\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.73176, size = 41, normalized size = 1.21 \begin{align*} \begin{cases} \frac{x \operatorname{acoth}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{4 b} - \frac{\operatorname{acoth}^{5}{\left (\tanh{\left (a + b x \right )} \right )}}{20 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \operatorname{acoth}^{3}{\left (\tanh{\left (a \right )} \right )}}{2} & \text{otherwise} \end{cases} \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 x \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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