Optimal. Leaf size=14 \[ -\frac{1}{2} a c x^2-c x \]
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Rubi [C] time = 0.0120579, antiderivative size = 26, normalized size of antiderivative = 1.86, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2288} \[ \frac{c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a} \]
Antiderivative was successfully verified.
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Rule 2288
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx &=\frac{c e^{2 \coth ^{-1}(a x)} \left (1-a^2 x^2\right )}{2 a}\\ \end{align*}
Mathematica [C] time = 0.0080872, size = 26, normalized size = 1.86 \[ \frac{c \left (1-a^2 x^2\right ) e^{2 \coth ^{-1}(a x)}}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 13, normalized size = 0.9 \begin{align*} c \left ( -{\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 = 1.03664, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2} \, a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43824, size = 27, normalized size = 1.93 \begin{align*} -\frac{1}{2} \, a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.071802, size = 12, normalized size = 0.86 \begin{align*} - \frac{a c x^{2}}{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15182, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2} \, a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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