Optimal. Leaf size=16 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 b} \]
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Rubi [A] time = 0.0044888, antiderivative size = 16, 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 = {2157, 30} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 2157
Rule 30
Rubi steps
\begin{align*} \int \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0057057, size = 16, normalized size = 1. \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 15, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53072, size = 69, normalized size = 4.31 \begin{align*} -\frac{3}{2} \, b x^{2} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + x \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{4} \,{\left (b^{2} x^{4} - 4 \, b x^{3} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.41287, size = 66, normalized size = 4.12 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.0068, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{4 b} & \text{for}\: b \neq 0 \\x \operatorname{atanh}^{3}{\left (\tanh{\left (a \right )} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11, size = 42, normalized size = 2.62 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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