Optimal. Leaf size=71 \[ \frac{a^3}{c (1-a x)}-\frac{3 a^2}{c x}+\frac{4 a^3 \log (x)}{c}-\frac{4 a^3 \log (1-a x)}{c}-\frac{a}{c x^2}-\frac{1}{3 c x^3} \]
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Rubi [A] time = 0.109847, antiderivative size = 71, 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} \[ \frac{a^3}{c (1-a x)}-\frac{3 a^2}{c x}+\frac{4 a^3 \log (x)}{c}-\frac{4 a^3 \log (1-a x)}{c}-\frac{a}{c x^2}-\frac{1}{3 c x^3} \]
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^4 \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1}{x^4 (1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{x^4}+\frac{2 a}{x^3}+\frac{3 a^2}{x^2}+\frac{4 a^3}{x}+\frac{a^4}{(-1+a x)^2}-\frac{4 a^4}{-1+a x}\right ) \, dx}{c}\\ &=-\frac{1}{3 c x^3}-\frac{a}{c x^2}-\frac{3 a^2}{c x}+\frac{a^3}{c (1-a x)}+\frac{4 a^3 \log (x)}{c}-\frac{4 a^3 \log (1-a x)}{c}\\ \end{align*}
Mathematica [A] time = 0.0393452, size = 71, normalized size = 1. \[ \frac{a^3}{c (1-a x)}-\frac{3 a^2}{c x}+\frac{4 a^3 \log (x)}{c}-\frac{4 a^3 \log (1-a x)}{c}-\frac{a}{c x^2}-\frac{1}{3 c x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 69, normalized size = 1. \begin{align*} -{\frac{1}{3\,c{x}^{3}}}-{\frac{a}{c{x}^{2}}}-3\,{\frac{{a}^{2}}{cx}}+4\,{\frac{{a}^{3}\ln \left ( x \right ) }{c}}-{\frac{{a}^{3}}{c \left ( ax-1 \right ) }}-4\,{\frac{{a}^{3}\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 = 0.94526, size = 86, normalized size = 1.21 \begin{align*} -\frac{4 \, a^{3} \log \left (a x - 1\right )}{c} + \frac{4 \, a^{3} \log \left (x\right )}{c} - \frac{12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x - 1}{3 \,{\left (a c x^{4} - c x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04781, size = 177, normalized size = 2.49 \begin{align*} -\frac{12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x + 12 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 12 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 1}{3 \,{\left (a c x^{4} - c x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.492618, size = 54, normalized size = 0.76 \begin{align*} \frac{4 a^{3} \left (\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}\right )}{c} - \frac{12 a^{3} x^{3} - 6 a^{2} x^{2} - 2 a x - 1}{3 a c x^{4} - 3 c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15145, size = 86, normalized size = 1.21 \begin{align*} -\frac{4 \, a^{3} \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac{4 \, a^{3} \log \left ({\left | x \right |}\right )}{c} - \frac{12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x - 1}{3 \,{\left (a x - 1\right )} c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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