Optimal. Leaf size=52 \[ \frac{1}{2 a c^3 (1-a x)^2}-\frac{4}{3 a c^3 (1-a x)^3}+\frac{1}{a c^3 (1-a x)^4} \]
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Rubi [A] time = 0.0680686, antiderivative size = 52, 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} \[ \frac{1}{2 a c^3 (1-a x)^2}-\frac{4}{3 a c^3 (1-a x)^3}+\frac{1}{a c^3 (1-a x)^4} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=\frac{\int \frac{(1+a x)^2}{(1-a x)^5} \, dx}{c^3}\\ &=\frac{\int \left (-\frac{4}{(-1+a x)^5}-\frac{4}{(-1+a x)^4}-\frac{1}{(-1+a x)^3}\right ) \, dx}{c^3}\\ &=\frac{1}{a c^3 (1-a x)^4}-\frac{4}{3 a c^3 (1-a x)^3}+\frac{1}{2 a c^3 (1-a x)^2}\\ \end{align*}
Mathematica [A] time = 0.0183849, size = 31, normalized size = 0.6 \[ \frac{3 a^2 x^2+2 a x+1}{6 a c^3 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 41, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{4}{3\,a \left ( ax-1 \right ) ^{3}}}+{\frac{1}{a \left ( ax-1 \right ) ^{4}}}+{\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 = 1.02431, size = 88, normalized size = 1.69 \begin{align*} \frac{3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, 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 = 1.52557, size = 131, normalized size = 2.52 \begin{align*} \frac{3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, 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 = 0.830525, size = 66, normalized size = 1.27 \begin{align*} \frac{3 a^{2} x^{2} + 2 a x + 1}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 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 = 1.15269, size = 57, normalized size = 1.1 \begin{align*} \frac{\frac{3}{{\left (a x - 1\right )}^{2} a} + \frac{8}{{\left (a x - 1\right )}^{3} a} + \frac{6}{{\left (a x - 1\right )}^{4} a}}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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