Optimal. Leaf size=37 \[ \frac{1}{2 a c^3 (1-a x)^2}-\frac{2}{3 a c^3 (1-a x)^3} \]
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Rubi [A] time = 0.0623262, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ \frac{1}{2 a c^3 (1-a x)^2}-\frac{2}{3 a c^3 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=-\frac{\int \frac{1+a x}{(1-a x)^4} \, dx}{c^3}\\ &=-\frac{\int \left (\frac{2}{(-1+a x)^4}+\frac{1}{(-1+a x)^3}\right ) \, dx}{c^3}\\ &=-\frac{2}{3 a c^3 (1-a x)^3}+\frac{1}{2 a c^3 (1-a x)^2}\\ \end{align*}
Mathematica [A] time = 0.013904, size = 23, normalized size = 0.62 \[ \frac{3 a x+1}{6 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{2}{3\,a \left ( ax-1 \right ) ^{3}}}+{\frac{1}{2\,a \left ( ax-1 \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988694, size = 63, normalized size = 1.7 \begin{align*} \frac{3 \, a x + 1}{6 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45622, size = 93, normalized size = 2.51 \begin{align*} \frac{3 \, a x + 1}{6 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.52817, size = 46, normalized size = 1.24 \begin{align*} \frac{3 a x + 1}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12439, size = 28, normalized size = 0.76 \begin{align*} \frac{3 \, a x + 1}{6 \,{\left (a x - 1\right )}^{3} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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