Optimal. Leaf size=87 \[ \frac{14}{a c^4 (1-a x)}-\frac{8}{a c^4 (1-a x)^2}+\frac{3}{a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{6 \log (1-a x)}{a c^4}+\frac{x}{c^4} \]
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Rubi [A] time = 0.172946, antiderivative size = 87, 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} \[ \frac{14}{a c^4 (1-a x)}-\frac{8}{a c^4 (1-a x)^2}+\frac{3}{a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{6 \log (1-a x)}{a c^4}+\frac{x}{c^4} \]
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 )^4} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx\\ &=-\frac{a^4 \int \frac{e^{2 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=-\frac{a^4 \int \frac{x^4 (1+a x)}{(1-a x)^5} \, dx}{c^4}\\ &=-\frac{a^4 \int \left (-\frac{1}{a^4}-\frac{2}{a^4 (-1+a x)^5}-\frac{9}{a^4 (-1+a x)^4}-\frac{16}{a^4 (-1+a x)^3}-\frac{14}{a^4 (-1+a x)^2}-\frac{6}{a^4 (-1+a x)}\right ) \, dx}{c^4}\\ &=\frac{x}{c^4}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{3}{a c^4 (1-a x)^3}-\frac{8}{a c^4 (1-a x)^2}+\frac{14}{a c^4 (1-a x)}+\frac{6 \log (1-a x)}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.149446, size = 71, normalized size = 0.82 \[ \frac{2 a^5 x^5-8 a^4 x^4-16 a^3 x^3+60 a^2 x^2-56 a x+12 (a x-1)^4 \log (1-a x)+17}{2 a c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 81, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{4}}}-8\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{2}}}-3\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{3}}}+6\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{4}}}-{\frac{1}{2\,a{c}^{4} \left ( ax-1 \right ) ^{4}}}-14\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02608, size = 126, normalized size = 1.45 \begin{align*} -\frac{28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac{x}{c^{4}} + \frac{6 \, \log \left (a x - 1\right )}{a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55305, size = 271, normalized size = 3.11 \begin{align*} \frac{2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 56 \, a x + 12 \,{\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 ) + 17}{2 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.648879, size = 94, normalized size = 1.08 \begin{align*} - \frac{28 a^{3} x^{3} - 68 a^{2} x^{2} + 58 a x - 17}{2 a^{5} c^{4} x^{4} - 8 a^{4} c^{4} x^{3} + 12 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 2 a c^{4}} + \frac{x}{c^{4}} + \frac{6 \log{\left (a x - 1 \right )}}{a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12609, size = 78, normalized size = 0.9 \begin{align*} \frac{x}{c^{4}} + \frac{6 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{4}} - \frac{28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \,{\left (a x - 1\right )}^{4} a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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