Optimal. Leaf size=46 \[ -\frac{c (a x+1)^3}{3 a}-\frac{c (a x+1)^2}{a}-\frac{8 c \log (1-a x)}{a}-4 c x \]
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Rubi [A] time = 0.0477622, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6167, 6140, 43} \[ -\frac{c (a x+1)^3}{3 a}-\frac{c (a x+1)^2}{a}-\frac{8 c \log (1-a x)}{a}-4 c x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx\\ &=c \int \frac{(1+a x)^3}{1-a x} \, dx\\ &=c \int \left (-4+\frac{8}{1-a x}-2 (1+a x)-(1+a x)^2\right ) \, dx\\ &=-4 c x-\frac{c (1+a x)^2}{a}-\frac{c (1+a x)^3}{3 a}-\frac{8 c \log (1-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0137819, size = 36, normalized size = 0.78 \[ -\frac{1}{3} a^2 c x^3-2 a c x^2-\frac{8 c \log (1-a x)}{a}-7 c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 34, normalized size = 0.7 \begin{align*} -{\frac{c{x}^{3}{a}^{2}}{3}}-2\,ac{x}^{2}-7\,cx-8\,{\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 = 1.09658, size = 45, normalized size = 0.98 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} - 2 \, a c x^{2} - 7 \, c x - \frac{8 \, 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 = 1.63154, size = 88, normalized size = 1.91 \begin{align*} -\frac{a^{3} c x^{3} + 6 \, a^{2} c x^{2} + 21 \, a c x + 24 \, c \log \left (a x - 1\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.293501, size = 36, normalized size = 0.78 \begin{align*} - \frac{a^{2} c x^{3}}{3} - 2 a c x^{2} - 7 c x - \frac{8 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 = 1.12558, size = 81, normalized size = 1.76 \begin{align*} -\frac{{\left (a x - 1\right )}^{3}{\left (c + \frac{9 \, c}{a x - 1} + \frac{36 \, c}{{\left (a x - 1\right )}^{2}}\right )}}{3 \, a} + \frac{8 \, 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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