Optimal. Leaf size=23 \[ \frac{1}{2} x^2 \coth ^{-1}(\coth (a+b x))-\frac{b x^3}{6} \]
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Rubi [A] time = 0.0077072, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6242, 30} \[ \frac{1}{2} x^2 \coth ^{-1}(\coth (a+b x))-\frac{b x^3}{6} \]
Antiderivative was successfully verified.
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Rule 6242
Rule 30
Rubi steps
\begin{align*} \int x \coth ^{-1}(\coth (a+b x)) \, dx &=\frac{1}{2} x^2 \coth ^{-1}(\coth (a+b x))-\frac{1}{2} b \int x^2 \, dx\\ &=-\frac{b x^3}{6}+\frac{1}{2} x^2 \coth ^{-1}(\coth (a+b x))\\ \end{align*}
Mathematica [A] time = 0.015436, size = 20, normalized size = 0.87 \[ -\frac{1}{6} x^2 \left (b x-3 \coth ^{-1}(\coth (a+b x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 48, normalized size = 2.1 \begin{align*}{\frac{{x}^{2}{\rm arccoth} \left ({\rm coth} \left (bx+a\right )\right )}{2}}+{\frac{1}{2\,{b}^{2}} \left ( -{\frac{ \left ( bx+a \right ) ^{3}}{3}}+ \left ( bx+a \right ) ^{2}a-{a}^{2} \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.933736, size = 18, normalized size = 0.78 \begin{align*} \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26847, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{3} x^{3} b + \frac{1}{2} x^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.9831, size = 60, normalized size = 2.61 \begin{align*} \begin{cases} 0 & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\\frac{x^{2} \operatorname{acoth}{\left (\coth{\left (a \right )} \right )}}{2} & \text{for}\: b = 0 \\\frac{x \operatorname{acoth}^{2}{\left (\frac{1}{\tanh{\left (a + b x \right )}} \right )}}{2 b} - \frac{\operatorname{acoth}^{3}{\left (\frac{1}{\tanh{\left (a + b x \right )}} \right )}}{6 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13859, size = 18, normalized size = 0.78 \begin{align*} \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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