Optimal. Leaf size=26 \[ -\frac{1}{2} a c x^2-\frac{4 c \log (a x+1)}{a}+3 c x \]
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Rubi [A] time = 0.0350002, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6167, 6129, 43} \[ -\frac{1}{2} a c x^2-\frac{4 c \log (a x+1)}{a}+3 c x \]
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) \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x) \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{1+a x} \, dx\right )\\ &=-\left (c \int \left (-3+a x+\frac{4}{1+a x}\right ) \, dx\right )\\ &=3 c x-\frac{1}{2} a c x^2-\frac{4 c \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0092851, size = 26, normalized size = 1. \[ -\frac{1}{2} a c x^2-\frac{4 c \log (a x+1)}{a}+3 c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 1. \begin{align*} 3\,cx-{\frac{ac{x}^{2}}{2}}-4\,{\frac{c\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02372, size = 32, normalized size = 1.23 \begin{align*} -\frac{1}{2} \, a c x^{2} + 3 \, c x - \frac{4 \, c \log \left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46784, size = 66, normalized size = 2.54 \begin{align*} -\frac{a^{2} c x^{2} - 6 \, a c x + 8 \, c \log \left (a x + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09001, size = 24, normalized size = 0.92 \begin{align*} - \frac{a c x^{2}}{2} + 3 c x - \frac{4 c \log{\left (a x + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12306, size = 47, normalized size = 1.81 \begin{align*} -\frac{4 \, c \log \left ({\left | a x + 1 \right |}\right )}{a} - \frac{a^{3} c x^{2} - 6 \, a^{2} c x}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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