Optimal. Leaf size=42 \[ -\frac{1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^5}{30} \]
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Rubi [A] time = 0.0221278, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac{1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^5}{30} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^2 \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{3} (2 b) \int x^3 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{6} b^2 \int x^4 \, dx\\ &=\frac{b^2 x^5}{30}-\frac{1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0478986, size = 37, normalized size = 0.88 \[ \frac{1}{30} x^3 \left (-5 b x \tanh ^{-1}(\tanh (a+b x))+10 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 38, normalized size = 0.9 \begin{align*}{\frac{{x}^{3} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{3}}-{\frac{2\,b}{3} \left ({\frac{{x}^{4}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{4}}-{\frac{b{x}^{5}}{20}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34477, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{30} \, b^{2} x^{5} - \frac{1}{6} \, b x^{4} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) + \frac{1}{3} \, x^{3} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53462, size = 55, normalized size = 1.31 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.74069, size = 37, normalized size = 0.88 \begin{align*} \frac{b^{2} x^{5}}{30} - \frac{b x^{4} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{6} + \frac{x^{3} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14106, size = 32, normalized size = 0.76 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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