Optimal. Leaf size=20 \[ \frac{1}{3} a^2 c^2 x^3-c^2 x \]
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Rubi [A] time = 0.0482971, antiderivative size = 20, 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, 41} \[ \frac{1}{3} a^2 c^2 x^3-c^2 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 41
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\\ &=-\left (c^2 \int (1-a x) (1+a x) \, dx\right )\\ &=-\left (c^2 \int \left (1-a^2 x^2\right ) \, dx\right )\\ &=-c^2 x+\frac{1}{3} a^2 c^2 x^3\\ \end{align*}
Mathematica [A] time = 0.0077956, size = 17, normalized size = 0.85 \[ -c^2 \left (x-\frac{a^2 x^3}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 17, normalized size = 0.9 \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 = 1.06875, size = 24, normalized size = 1.2 \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 = 1.41356, size = 34, normalized size = 1.7 \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 = 0.080905, size = 15, normalized size = 0.75 \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 = 1.13679, size = 24, normalized size = 1.2 \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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