Optimal. Leaf size=19 \[ c^2 x-\frac{1}{3} a^2 c^2 x^3 \]
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Rubi [A] time = 0.0255281, antiderivative size = 19, 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, 41} \[ c^2 x-\frac{1}{3} a^2 c^2 x^3 \]
Antiderivative was successfully verified.
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Rule 6129
Rule 41
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c^2 \int (1-a x) (1+a x) \, dx\\ &=c^2 \int \left (1-a^2 x^2\right ) \, dx\\ &=c^2 x-\frac{1}{3} a^2 c^2 x^3\\ \end{align*}
Mathematica [A] time = 0.0082351, size = 16, normalized size = 0.84 \[ c^2 \left (x-\frac{a^2 x^3}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 15, normalized size = 0.8 \begin{align*}{c}^{2} \left ( -{\frac{{x}^{3}{a}^{2}}{3}}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959621, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{3} \, a^{2} c^{2} x^{3} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54131, size = 35, normalized size = 1.84 \begin{align*} -\frac{1}{3} \, a^{2} c^{2} x^{3} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.085084, size = 15, normalized size = 0.79 \begin{align*} - \frac{a^{2} c^{2} x^{3}}{3} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23071, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{3} \, a^{2} c^{2} x^{3} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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