Optimal. Leaf size=53 \[ \frac{5}{a c^2 (1-a x)}-\frac{1}{a c^2 (1-a x)^2}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]
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Rubi [A] time = 0.15467, antiderivative size = 53, 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} \[ \frac{5}{a c^2 (1-a x)}-\frac{1}{a c^2 (1-a x)^2}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]
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 )^2} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx\\ &=-\frac{a^2 \int \frac{e^{2 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=-\frac{a^2 \int \frac{x^2 (1+a x)}{(1-a x)^3} \, dx}{c^2}\\ &=-\frac{a^2 \int \left (-\frac{1}{a^2}-\frac{2}{a^2 (-1+a x)^3}-\frac{5}{a^2 (-1+a x)^2}-\frac{4}{a^2 (-1+a x)}\right ) \, dx}{c^2}\\ &=\frac{x}{c^2}-\frac{1}{a c^2 (1-a x)^2}+\frac{5}{a c^2 (1-a x)}+\frac{4 \log (1-a x)}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.09914, size = 51, normalized size = 0.96 \[ -\frac{5}{a c^2 (a x-1)}-\frac{1}{a c^2 (a x-1)^2}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 51, normalized size = 1. \begin{align*}{\frac{x}{{c}^{2}}}-{\frac{1}{a{c}^{2} \left ( ax-1 \right ) ^{2}}}+4\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{2}}}-5\,{\frac{1}{a{c}^{2} \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04903, size = 74, normalized size = 1.4 \begin{align*} -\frac{5 \, a x - 4}{a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}} + \frac{x}{c^{2}} + \frac{4 \, \log \left (a x - 1\right )}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5058, size = 149, normalized size = 2.81 \begin{align*} \frac{a^{3} x^{3} - 2 \, a^{2} x^{2} - 4 \, a x + 4 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 4}{a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.435774, size = 49, normalized size = 0.92 \begin{align*} - \frac{5 a x - 4}{a^{3} c^{2} x^{2} - 2 a^{2} c^{2} x + a c^{2}} + \frac{x}{c^{2}} + \frac{4 \log{\left (a x - 1 \right )}}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14232, size = 57, normalized size = 1.08 \begin{align*} \frac{x}{c^{2}} + \frac{4 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{2}} - \frac{5 \, a x - 4}{{\left (a x - 1\right )}^{2} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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