Optimal. Leaf size=37 \[ \frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a} \]
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Rubi [A] time = 0.0347904, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int (1-a x)^2 (1+a x) \, dx\\ &=c^3 \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx\\ &=-\frac{2 c^3 (1-a x)^3}{3 a}+\frac{c^3 (1-a x)^4}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0135797, size = 30, normalized size = 0.81 \[ \frac{1}{12} c^3 x \left (3 a^3 x^3-4 a^2 x^2-6 a x+12\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 29, normalized size = 0.8 \begin{align*}{c}^{3} \left ({\frac{{x}^{4}{a}^{3}}{4}}-{\frac{{x}^{3}{a}^{2}}{3}}-{\frac{a{x}^{2}}{2}}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957431, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{4} \, a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} - \frac{1}{2} \, a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56118, size = 80, normalized size = 2.16 \begin{align*} \frac{1}{4} \, a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} - \frac{1}{2} \, a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.089358, size = 37, normalized size = 1. \begin{align*} \frac{a^{3} c^{3} x^{4}}{4} - \frac{a^{2} c^{3} x^{3}}{3} - \frac{a c^{3} x^{2}}{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18426, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{4} \, a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} - \frac{1}{2} \, a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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