Optimal. Leaf size=30 \[ \frac{1}{a^2 c (1-a x)}+\frac{\log (1-a x)}{a^2 c} \]
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Rubi [A] time = 0.06539, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6150, 43} \[ \frac{1}{a^2 c (1-a x)}+\frac{\log (1-a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x}{(1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{a (-1+a x)^2}+\frac{1}{a (-1+a x)}\right ) \, dx}{c}\\ &=\frac{1}{a^2 c (1-a x)}+\frac{\log (1-a x)}{a^2 c}\\ \end{align*}
Mathematica [A] time = 0.0188966, size = 23, normalized size = 0.77 \[ \frac{\frac{1}{1-a x}+\log (1-a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 30, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{2}c \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{{a}^{2}c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953644, size = 42, normalized size = 1.4 \begin{align*} -\frac{1}{a^{3} c x - a^{2} c} + \frac{\log \left (a x - 1\right )}{a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92952, size = 65, normalized size = 2.17 \begin{align*} \frac{{\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{a^{3} c x - a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.302677, size = 24, normalized size = 0.8 \begin{align*} - \frac{1}{a^{3} c x - a^{2} c} + \frac{\log{\left (a x - 1 \right )}}{a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13642, size = 41, normalized size = 1.37 \begin{align*} \frac{\log \left ({\left | a x - 1 \right |}\right )}{a^{2} c} - \frac{1}{{\left (a x - 1\right )} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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