Optimal. Leaf size=48 \[ -\frac{4}{a c (1-a x)}+\frac{2}{a c (1-a x)^2}-\frac{\log (1-a x)}{a c} \]
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Rubi [A] time = 0.0449021, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 43} \[ -\frac{4}{a c (1-a x)}+\frac{2}{a c (1-a x)^2}-\frac{\log (1-a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 \tanh ^{-1}(a x)}}{c-a c x} \, dx &=\frac{\int \frac{(1+a x)^2}{(1-a x)^3} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{1-a x}-\frac{4}{(-1+a x)^3}-\frac{4}{(-1+a x)^2}\right ) \, dx}{c}\\ &=\frac{2}{a c (1-a x)^2}-\frac{4}{a c (1-a x)}-\frac{\log (1-a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0184713, size = 36, normalized size = 0.75 \[ \frac{4 a x+(a x-1)^2 (-\log (1-a x))-2}{a c (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 46, normalized size = 1. \begin{align*} 4\,{\frac{1}{ac \left ( ax-1 \right ) }}+2\,{\frac{1}{ac \left ( ax-1 \right ) ^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952757, size = 59, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (2 \, a x - 1\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} - \frac{\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 = 1.64862, size = 108, normalized size = 2.25 \begin{align*} \frac{4 \, a x -{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 2}{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 = 0.435039, size = 36, normalized size = 0.75 \begin{align*} \frac{4 a x - 2}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac{\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 = 1.18496, size = 50, normalized size = 1.04 \begin{align*} -\frac{\log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac{2 \,{\left (2 \, a x - 1\right )}}{{\left (a x - 1\right )}^{2} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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