Optimal. Leaf size=25 \[ \frac{(a x+1)^3}{6 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.0504695, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 37} \[ \frac{(a x+1)^3}{6 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\\ &=\frac{\int \frac{(1+a x)^2}{(1-a x)^4} \, dx}{c^2}\\ &=\frac{(1+a x)^3}{6 a c^2 (1-a x)^3}\\ \end{align*}
Mathematica [A] time = 0.0089624, size = 25, normalized size = 1. \[ \frac{(a x+1)^3}{6 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 42, normalized size = 1.7 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{4}{3\,a \left ( ax-1 \right ) ^{3}}}-2\,{\frac{1}{a \left ( ax-1 \right ) ^{2}}}-{\frac{1}{a \left ( ax-1 \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02094, size = 69, normalized size = 2.76 \begin{align*} -\frac{3 \, a^{2} x^{2} + 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 = 1.43907, size = 100, normalized size = 4. \begin{align*} -\frac{3 \, a^{2} x^{2} + 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 = 1.1261, size = 51, normalized size = 2.04 \begin{align*} - \frac{3 a^{2} x^{2} + 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 [B] time = 1.15261, size = 68, normalized size = 2.72 \begin{align*} -\frac{2}{{\left (a c x - c\right )}^{2} a} - \frac{1}{{\left (a c x - c\right )} a c} - \frac{4 \, c}{3 \,{\left (a c x - c\right )}^{3} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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