Optimal. Leaf size=53 \[ \frac{8}{a c (1-a x)}-\frac{2}{a c (1-a x)^2}+\frac{5 \log (1-a x)}{a c}+\frac{x}{c} \]
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Rubi [A] time = 0.133421, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 77} \[ \frac{8}{a c (1-a x)}-\frac{2}{a c (1-a x)^2}+\frac{5 \log (1-a x)}{a c}+\frac{x}{c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx\\ &=-\frac{a \int \frac{e^{4 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac{a \int \frac{x (1+a x)^2}{(1-a x)^3} \, dx}{c}\\ &=-\frac{a \int \left (-\frac{1}{a}-\frac{4}{a (-1+a x)^3}-\frac{8}{a (-1+a x)^2}-\frac{5}{a (-1+a x)}\right ) \, dx}{c}\\ &=\frac{x}{c}-\frac{2}{a c (1-a x)^2}+\frac{8}{a c (1-a x)}+\frac{5 \log (1-a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0304802, size = 51, normalized size = 0.96 \[ -\frac{a \left (-\frac{8}{a^2 (1-a x)}+\frac{2}{a^2 (1-a x)^2}-\frac{5 \log (1-a x)}{a^2}-\frac{x}{a}\right )}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 51, normalized size = 1. \begin{align*}{\frac{x}{c}}-2\,{\frac{1}{ac \left ( ax-1 \right ) ^{2}}}+5\,{\frac{\ln \left ( ax-1 \right ) }{ac}}-8\,{\frac{1}{ac \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03053, size = 66, normalized size = 1.25 \begin{align*} -\frac{2 \,{\left (4 \, a x - 3\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} + \frac{x}{c} + \frac{5 \, \log \left (a x - 1\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43967, size = 140, normalized size = 2.64 \begin{align*} \frac{a^{3} x^{3} - 2 \, a^{2} x^{2} - 7 \, a x + 5 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 6}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.429769, size = 41, normalized size = 0.77 \begin{align*} - \frac{8 a x - 6}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{x}{c} + \frac{5 \log{\left (a x - 1 \right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11963, size = 100, normalized size = 1.89 \begin{align*} \frac{a x - 1}{a c} - \frac{5 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c} - \frac{2 \,{\left (\frac{4 \, a^{3} c}{a x - 1} + \frac{a^{3} c}{{\left (a x - 1\right )}^{2}}\right )}}{a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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