Optimal. Leaf size=23 \[ \frac{1}{2} x^2 \tanh ^{-1}(\tanh (a+b x))-\frac{b x^3}{6} \]
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Rubi [A] time = 0.0068111, 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 = {6239, 30} \[ \frac{1}{2} x^2 \tanh ^{-1}(\tanh (a+b x))-\frac{b x^3}{6} \]
Antiderivative was successfully verified.
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Rule 6239
Rule 30
Rubi steps
\begin{align*} \int x \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac{1}{2} x^2 \tanh ^{-1}(\tanh (a+b x))-\frac{1}{2} b \int x^2 \, dx\\ &=-\frac{b x^3}{6}+\frac{1}{2} x^2 \tanh ^{-1}(\tanh (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0150803, size = 20, normalized size = 0.87 \[ -\frac{1}{6} x^2 \left (b x-3 \tanh ^{-1}(\tanh (a+b x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{3}}{6}}+{\frac{{x}^{2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14261, size = 26, normalized size = 1.13 \begin{align*} -\frac{1}{6} \, b x^{3} + \frac{1}{2} \, x^{2} \operatorname{artanh}\left (\tanh \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 = 1.61921, size = 31, normalized size = 1.35 \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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Sympy [A] time = 3.21176, size = 39, normalized size = 1.7 \begin{align*} \begin{cases} \frac{x \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{2 b} - \frac{\operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{6 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \operatorname{atanh}{\left (\tanh{\left (a \right )} \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12741, 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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