Optimal. Leaf size=27 \[ -\frac{1}{2} a c x^2-\frac{4 c \log (1-a x)}{a}-3 c x \]
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Rubi [A] time = 0.0366372, antiderivative size = 27, 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} \[ -\frac{1}{2} a c x^2-\frac{4 c \log (1-a x)}{a}-3 c x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x) \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x) \, dx\\ &=c \int \frac{(1+a x)^2}{1-a x} \, dx\\ &=c \int \left (-3-a x+\frac{4}{1-a x}\right ) \, dx\\ &=-3 c x-\frac{1}{2} a c x^2-\frac{4 c \log (1-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0122167, size = 26, normalized size = 0.96 \[ c \left (-\frac{a x^2}{2}-\frac{4 \log (1-a x)}{a}-3 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 0.9 \begin{align*} -{\frac{ac{x}^{2}}{2}}-3\,cx-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 = 1.02561, size = 32, normalized size = 1.19 \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 = 1.50781, size = 66, normalized size = 2.44 \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 = 0.470965, size = 26, normalized size = 0.96 \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 = 1.17871, size = 68, normalized size = 2.52 \begin{align*} -\frac{{\left (a x - 1\right )}^{2}{\left (c + \frac{8 \, c}{a x - 1}\right )}}{2 \, a} + \frac{4 \, c \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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