Optimal. Leaf size=88 \[ -\frac{14}{a c^4 (1-a x)}+\frac{8}{a c^4 (1-a x)^2}-\frac{3}{a c^4 (1-a x)^3}+\frac{1}{2 a c^4 (1-a x)^4}-\frac{6 \log (1-a x)}{a c^4}-\frac{x}{c^4} \]
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Rubi [A] time = 0.138521, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {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 6131
Rule 6129
Rule 77
Rubi steps
\begin{align*} \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 = 0.147132, size = 71, normalized size = 0.81 \[ \frac{-2 a^5 x^5+8 a^4 x^4+16 a^3 x^3-60 a^2 x^2+56 a x-12 (a x-1)^4 \log (1-a x)-17}{2 a c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 82, normalized size = 0.9 \begin{align*} -{\frac{x}{{c}^{4}}}+3\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{3}}}+8\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{2}}}+14\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) }}-6\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{4}}}+{\frac{1}{2\,a{c}^{4} \left ( ax-1 \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95665, size = 127, normalized size = 1.44 \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 = 2.18946, size = 273, normalized size = 3.1 \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 = 0.651929, size = 94, normalized size = 1.07 \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 = 1.13769, size = 80, normalized size = 0.91 \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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