Optimal. Leaf size=61 \[ \frac{1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac{1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{280} b^3 x^8 \]
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Rubi [A] time = 0.0411641, antiderivative size = 61, 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, 30} \[ \frac{1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac{1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{280} b^3 x^8 \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^4 \coth ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{5} (3 b) \int x^5 \coth ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3+\frac{1}{5} b^2 \int x^6 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac{1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{35} b^3 \int x^7 \, dx\\ &=-\frac{1}{280} b^3 x^8+\frac{1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac{1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3\\ \end{align*}
Mathematica [A] time = 0.030511, size = 54, normalized size = 0.89 \[ -\frac{1}{280} x^5 \left (-8 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+28 b x \coth ^{-1}(\tanh (a+b x))^2-56 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 1.102, size = 18111, normalized size = 296.9 \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 = 1.54533, size = 73, normalized size = 1.2 \begin{align*} -\frac{1}{10} \, b x^{6} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{1}{5} \, x^{5} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{280} \,{\left (b^{2} x^{8} - 8 \, b x^{7} \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 = 1.59236, size = 119, normalized size = 1.95 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{7} \, a b^{2} x^{7} - \frac{1}{8} \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{6} - \frac{1}{20} \,{\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.05556, size = 56, normalized size = 0.92 \begin{align*} - \frac{b^{3} x^{8}}{280} + \frac{b^{2} x^{7} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{35} - \frac{b x^{6} \operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{10} + \frac{x^{5} \operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{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 x^{4} \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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