Optimal. Leaf size=89 \[ \frac{19}{a c^3 (1-a x)}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{7 \log (1-a x)}{a c^3}+\frac{x}{c^3} \]
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Rubi [A] time = 0.172051, antiderivative size = 89, 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, 88} \[ \frac{19}{a c^3 (1-a x)}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{7 \log (1-a x)}{a c^3}+\frac{x}{c^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx\\ &=-\frac{a^3 \int \frac{e^{4 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac{a^3 \int \frac{x^3 (1+a x)^2}{(1-a x)^5} \, dx}{c^3}\\ &=-\frac{a^3 \int \left (-\frac{1}{a^3}-\frac{4}{a^3 (-1+a x)^5}-\frac{16}{a^3 (-1+a x)^4}-\frac{25}{a^3 (-1+a x)^3}-\frac{19}{a^3 (-1+a x)^2}-\frac{7}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=\frac{x}{c^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{19}{a c^3 (1-a x)}+\frac{7 \log (1-a x)}{a c^3}\\ \end{align*}
Mathematica [A] time = 0.139355, size = 71, normalized size = 0.8 \[ \frac{6 a^5 x^5-24 a^4 x^4-78 a^3 x^3+243 a^2 x^2-218 a x+42 (a x-1)^4 \log (1-a x)+65}{6 a c^3 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 81, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{3}}}-{\frac{25}{2\,a{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{16}{3\,a{c}^{3} \left ( ax-1 \right ) ^{3}}}+7\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{3}}}-{\frac{1}{a{c}^{3} \left ( ax-1 \right ) ^{4}}}-19\,{\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 = 1.04203, size = 126, normalized size = 1.42 \begin{align*} -\frac{114 \, a^{3} x^{3} - 267 \, a^{2} x^{2} + 224 \, a x - 65}{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 )}} + \frac{x}{c^{3}} + \frac{7 \, \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 = 1.84076, size = 275, normalized size = 3.09 \begin{align*} \frac{6 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 78 \, a^{3} x^{3} + 243 \, a^{2} x^{2} - 218 \, a x + 42 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 65}{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 = 0.663173, size = 94, normalized size = 1.06 \begin{align*} - \frac{114 a^{3} x^{3} - 267 a^{2} x^{2} + 224 a x - 65}{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}} + \frac{x}{c^{3}} + \frac{7 \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 = 1.15071, size = 147, normalized size = 1.65 \begin{align*} \frac{a x - 1}{a c^{3}} - \frac{7 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c^{3}} - \frac{\frac{114 \, a^{7} c^{9}}{a x - 1} + \frac{75 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac{32 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac{6 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{4}}}{6 \, a^{8} c^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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