Optimal. Leaf size=37 \[ \frac{2 c^3 (1-a x)^3}{3 a}-\frac{c^3 (1-a x)^4}{4 a} \]
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Rubi [A] time = 0.0591795, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ \frac{2 c^3 (1-a x)^3}{3 a}-\frac{c^3 (1-a x)^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\\ &=-\left (c^3 \int (1-a x)^2 (1+a x) \, dx\right )\\ &=-\left (c^3 \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx\right )\\ &=\frac{2 c^3 (1-a x)^3}{3 a}-\frac{c^3 (1-a x)^4}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0133888, 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 = 0.046, size = 31, 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 = 1.00857, size = 51, normalized size = 1.38 \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 = 1.52194, size = 81, normalized size = 2.19 \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 = 0.084199, 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 = 1.22017, size = 51, normalized size = 1.38 \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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