Optimal. Leaf size=75 \[ \frac{3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{3 x^2}{2 b^2} \]
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Rubi [A] time = 0.0537226, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2168, 2159, 2158, 2157, 29} \[ \frac{3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{3 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2158
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{x^3}{\coth ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{3 \int \frac{x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{3 x^2}{2 b^2}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}-\frac{\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac{3 x^2}{2 b^2}+\frac{3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac{1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac{3 x^2}{2 b^2}+\frac{3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ &=\frac{3 x^2}{2 b^2}+\frac{3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0615903, size = 83, normalized size = 1.11 \[ \frac{\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3}{b^4 \coth ^{-1}(\tanh (a+b x))}-\frac{2 x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{b^3}+\frac{3 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac{x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.408, size = 29109, normalized size = 388.1 \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 [C] time = 2.43762, size = 167, normalized size = 2.23 \begin{align*} \frac{4 \,{\left (4 \, b^{3} x^{3} + i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3} +{\left (6 i \, \pi b^{2} - 12 \, a b^{2}\right )} x^{2} +{\left (4 \, \pi ^{2} b + 16 i \, \pi a b - 16 \, a^{2} b\right )} x\right )}}{32 \, b^{5} x - 16 i \, \pi b^{4} + 32 \, a b^{4}} - \frac{{\left (3 \, \pi ^{2} + 12 i \, \pi a - 12 \, a^{2}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76149, size = 544, normalized size = 7.25 \begin{align*} \frac{16 \, b^{4} x^{4} - 32 \, a b^{3} x^{3} - 2 \, \pi ^{4} + 32 \, a^{4} + 4 \,{\left (\pi ^{2} b^{2} - 28 \, a^{2} b^{2}\right )} x^{2} - 8 \,{\left (5 \, \pi ^{2} a b + 4 \, a^{3} b\right )} x - 48 \,{\left (4 \, \pi a b^{2} x^{2} + 8 \, \pi a^{2} b x + \pi ^{3} a + 4 \, \pi a^{3}\right )} \arctan \left (-\frac{2 \, b x + 2 \, a - \sqrt{4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) - 3 \,{\left (\pi ^{4} - 16 \, a^{4} + 4 \,{\left (\pi ^{2} b^{2} - 4 \, a^{2} b^{2}\right )} x^{2} + 8 \,{\left (\pi ^{2} a b - 4 \, a^{3} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{8 \,{\left (4 \, b^{6} x^{2} + 8 \, a b^{5} x + \pi ^{2} b^{4} + 4 \, a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \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{x^{3}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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