Optimal. Leaf size=23 \[ \frac{1}{3} x^3 \coth ^{-1}(\coth (a+b x))-\frac{b x^4}{12} \]
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Rubi [A] time = 0.014254, 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} \[ \frac{1}{3} x^3 \coth ^{-1}(\coth (a+b x))-\frac{b x^4}{12} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}(\coth (a+b x)) \, dx &=\frac{1}{3} x^3 \coth ^{-1}(\coth (a+b x))-\frac{1}{3} b \int x^3 \, dx\\ &=-\frac{b x^4}{12}+\frac{1}{3} x^3 \coth ^{-1}(\coth (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0213134, size = 20, normalized size = 0.87 \[ -\frac{1}{12} x^3 \left (b x-4 \coth ^{-1}(\coth (a+b x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.092, size = 59, normalized size = 2.6 \begin{align*}{\frac{{x}^{3}{\rm arccoth} \left ({\rm coth} \left (bx+a\right )\right )}{3}}+{\frac{1}{3\,{b}^{3}} \left ( -{\frac{ \left ( bx+a \right ) ^{4}}{4}}+ \left ( bx+a \right ) ^{3}a-{\frac{3\,{a}^{2} \left ( bx+a \right ) ^{2}}{2}}+ \left ( bx+a \right ){a}^{3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945767, size = 18, normalized size = 0.78 \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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Fricas [A] time = 1.34393, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{4} x^{4} b + \frac{1}{3} x^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 91.8494, size = 39, normalized size = 1.7 \begin{align*} \begin{cases} 0 & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\- \frac{b x^{4}}{12} + \frac{x^{3} \operatorname{acoth}{\left (\frac{1}{\tanh{\left (a + b x \right )}} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17349, size = 18, normalized size = 0.78 \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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