Optimal. Leaf size=23 \[ \frac{1}{3} x^3 \tanh ^{-1}(\coth (a+b x))-\frac{b x^4}{12} \]
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Rubi [A] time = 0.0196044, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2168, 30} \[ \frac{1}{3} x^3 \tanh ^{-1}(\coth (a+b x))-\frac{b x^4}{12} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^2 \tanh ^{-1}(\coth (a+b x)) \, dx &=\frac{1}{3} x^3 \tanh ^{-1}(\coth (a+b x))-\frac{1}{3} b \int x^3 \, dx\\ &=-\frac{b x^4}{12}+\frac{1}{3} x^3 \tanh ^{-1}(\coth (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0237691, size = 20, normalized size = 0.87 \[ -\frac{1}{12} x^3 \left (b x-4 \tanh ^{-1}(\coth (a+b x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{4}}{12}}+{\frac{{x}^{3}{\it Artanh} \left ({\rm coth} \left (bx+a\right ) \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13334, size = 26, normalized size = 1.13 \begin{align*} -\frac{1}{12} \, b x^{4} + \frac{1}{3} \, x^{3} \operatorname{artanh}\left (\coth \left (b x + a\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08463, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17625, size = 96, normalized size = 4.17 \begin{align*} -\frac{1}{12} \, b x^{4} + \frac{1}{6} \, x^{3} \log \left (-\frac{\frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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