Optimal. Leaf size=60 \[ \frac{a^2}{c (1-a x)}+\frac{3 a^2 \log (x)}{c}-\frac{3 a^2 \log (1-a x)}{c}-\frac{2 a}{c x}-\frac{1}{2 c x^2} \]
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Rubi [A] time = 0.101499, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 44} \[ \frac{a^2}{c (1-a x)}+\frac{3 a^2 \log (x)}{c}-\frac{3 a^2 \log (1-a x)}{c}-\frac{2 a}{c x}-\frac{1}{2 c x^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1}{x^3 (1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{x^3}+\frac{2 a}{x^2}+\frac{3 a^2}{x}+\frac{a^3}{(-1+a x)^2}-\frac{3 a^3}{-1+a x}\right ) \, dx}{c}\\ &=-\frac{1}{2 c x^2}-\frac{2 a}{c x}+\frac{a^2}{c (1-a x)}+\frac{3 a^2 \log (x)}{c}-\frac{3 a^2 \log (1-a x)}{c}\\ \end{align*}
Mathematica [A] time = 0.0477582, size = 52, normalized size = 0.87 \[ \frac{\frac{-6 a^2 x^2+3 a x+1}{x^2 (a x-1)}+6 a^2 \log (x)-6 a^2 \log (1-a x)}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 58, normalized size = 1. \begin{align*} -{\frac{1}{2\,c{x}^{2}}}-2\,{\frac{a}{cx}}+3\,{\frac{{a}^{2}\ln \left ( x \right ) }{c}}-{\frac{{a}^{2}}{c \left ( ax-1 \right ) }}-3\,{\frac{{a}^{2}\ln \left ( ax-1 \right ) }{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956291, size = 76, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{2} \log \left (a x - 1\right )}{c} + \frac{3 \, a^{2} \log \left (x\right )}{c} - \frac{6 \, a^{2} x^{2} - 3 \, a x - 1}{2 \,{\left (a c x^{3} - c x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03348, size = 157, normalized size = 2.62 \begin{align*} -\frac{6 \, a^{2} x^{2} - 3 \, a x + 6 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 6 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 1}{2 \,{\left (a c x^{3} - c x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.444763, size = 46, normalized size = 0.77 \begin{align*} \frac{3 a^{2} \left (\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}\right )}{c} - \frac{6 a^{2} x^{2} - 3 a x - 1}{2 a c x^{3} - 2 c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12318, size = 76, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{2} \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac{3 \, a^{2} \log \left ({\left | x \right |}\right )}{c} - \frac{6 \, a^{2} x^{2} - 3 \, a x - 1}{2 \,{\left (a x - 1\right )} c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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