Optimal. Leaf size=18 \[ \frac{1}{3 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.0597022, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 32} \[ \frac{1}{3 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\\ &=\frac{\int \frac{1}{(1-a x)^4} \, dx}{c^2}\\ &=\frac{1}{3 a c^2 (1-a x)^3}\\ \end{align*}
Mathematica [A] time = 0.0180535, size = 17, normalized size = 0.94 \[ -\frac{1}{3 a c^2 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{c}^{2} \left ( ax-1 \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01847, size = 55, normalized size = 3.06 \begin{align*} -\frac{1}{3 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39524, size = 78, normalized size = 4.33 \begin{align*} -\frac{1}{3 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.400754, size = 42, normalized size = 2.33 \begin{align*} - \frac{1}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13769, size = 20, normalized size = 1.11 \begin{align*} -\frac{1}{3 \,{\left (a x - 1\right )}^{3} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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