Optimal. Leaf size=73 \[ \frac{9}{a c^3 (1-a x)}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{2}{3 a c^3 (1-a x)^3}+\frac{5 \log (1-a x)}{a c^3}+\frac{x}{c^3} \]
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Rubi [A] time = 0.164493, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 77} \[ \frac{9}{a c^3 (1-a x)}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{2}{3 a c^3 (1-a x)^3}+\frac{5 \log (1-a x)}{a c^3}+\frac{x}{c^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx\\ &=\frac{a^3 \int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=\frac{a^3 \int \frac{x^3 (1+a x)}{(1-a x)^4} \, dx}{c^3}\\ &=\frac{a^3 \int \left (\frac{1}{a^3}+\frac{2}{a^3 (-1+a x)^4}+\frac{7}{a^3 (-1+a x)^3}+\frac{9}{a^3 (-1+a x)^2}+\frac{5}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=\frac{x}{c^3}+\frac{2}{3 a c^3 (1-a x)^3}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{9}{a c^3 (1-a x)}+\frac{5 \log (1-a x)}{a c^3}\\ \end{align*}
Mathematica [A] time = 0.120448, size = 63, normalized size = 0.86 \[ \frac{6 a^4 x^4-18 a^3 x^3-36 a^2 x^2+81 a x+30 (a x-1)^3 \log (1-a x)-37}{6 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 66, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{3}}}-{\frac{7}{2\,a{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{2}{3\,a{c}^{3} \left ( ax-1 \right ) ^{3}}}+5\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{3}}}-9\,{\frac{1}{a{c}^{3} \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04371, size = 101, normalized size = 1.38 \begin{align*} -\frac{54 \, a^{2} x^{2} - 87 \, a x + 37}{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 )}} + \frac{x}{c^{3}} + \frac{5 \, \log \left (a x - 1\right )}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50469, size = 217, normalized size = 2.97 \begin{align*} \frac{6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 37}{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 = 0.534018, size = 73, normalized size = 1. \begin{align*} - \frac{54 a^{2} x^{2} - 87 a x + 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} + \frac{x}{c^{3}} + \frac{5 \log{\left (a x - 1 \right )}}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12201, size = 68, normalized size = 0.93 \begin{align*} \frac{x}{c^{3}} + \frac{5 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{3}} - \frac{54 \, a^{2} x^{2} - 87 \, a x + 37}{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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