Optimal. Leaf size=53 \[ \frac{\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}-\frac{x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
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Rubi [A] time = 0.0312969, antiderivative size = 53, 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, 2157, 30} \[ \frac{\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}-\frac{x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2157
Rule 30
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\int x \coth ^{-1}(\tanh (a+b x))^4 \, dx}{2 b}\\ &=\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac{\int \coth ^{-1}(\tanh (a+b x))^5 \, dx}{10 b^2}\\ &=\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac{\operatorname{Subst}\left (\int x^5 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{10 b^3}\\ &=\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac{\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}\\ \end{align*}
Mathematica [A] time = 0.0222424, size = 54, normalized size = 1.02 \[ -\frac{1}{60} x^3 \left (-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+15 b x \coth ^{-1}(\tanh (a+b x))^2-20 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 1.081, size = 18111, normalized size = 341.7 \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 = 1.54597, size = 73, normalized size = 1.38 \begin{align*} -\frac{1}{4} \, b x^{4} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{1}{3} \, x^{3} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{60} \,{\left (b^{2} x^{6} - 6 \, b x^{5} \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 = 1.55735, size = 120, normalized size = 2.26 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{5} \, a b^{2} x^{5} - \frac{3}{16} \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{4} - \frac{1}{12} \,{\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.3897, size = 60, normalized size = 1.13 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acoth}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{4 b} - \frac{x \operatorname{acoth}^{5}{\left (\tanh{\left (a + b x \right )} \right )}}{10 b^{2}} + \frac{\operatorname{acoth}^{6}{\left (\tanh{\left (a + b x \right )} \right )}}{60 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \operatorname{acoth}^{3}{\left (\tanh{\left (a \right )} \right )}}{3} & \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^{2} \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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