Optimal. Leaf size=57 \[ \frac{1}{2 a c^3 (1-a x)}+\frac{5 \log (1-a x)}{4 a c^3}-\frac{\log (a x+1)}{4 a c^3}+\frac{x}{c^3} \]
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Rubi [A] time = 0.150284, antiderivative size = 57, 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} \[ \frac{1}{2 a c^3 (1-a x)}+\frac{5 \log (1-a x)}{4 a c^3}-\frac{\log (a x+1)}{4 a c^3}+\frac{x}{c^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-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)^2 (1+a x)} \, dx}{c^3}\\ &=\frac{a^3 \int \left (\frac{1}{a^3}+\frac{1}{2 a^3 (-1+a x)^2}+\frac{5}{4 a^3 (-1+a x)}-\frac{1}{4 a^3 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac{x}{c^3}+\frac{1}{2 a c^3 (1-a x)}+\frac{5 \log (1-a x)}{4 a c^3}-\frac{\log (1+a x)}{4 a c^3}\\ \end{align*}
Mathematica [A] time = 0.110389, size = 56, normalized size = 0.98 \[ -\frac{1}{2 a c^3 (a x-1)}+\frac{5 \log (1-a x)}{4 a c^3}-\frac{\log (a x+1)}{4 a c^3}+\frac{x}{c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 50, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{3}}}-{\frac{\ln \left ( ax+1 \right ) }{4\,a{c}^{3}}}-{\frac{1}{2\,a{c}^{3} \left ( ax-1 \right ) }}+{\frac{5\,\ln \left ( ax-1 \right ) }{4\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03942, size = 72, normalized size = 1.26 \begin{align*} -\frac{1}{2 \,{\left (a^{2} c^{3} x - a c^{3}\right )}} + \frac{x}{c^{3}} - \frac{\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac{5 \, \log \left (a x - 1\right )}{4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80426, size = 136, normalized size = 2.39 \begin{align*} \frac{4 \, a^{2} x^{2} - 4 \, a x -{\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (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 = 0.505827, size = 56, normalized size = 0.98 \begin{align*} a^{3} \left (- \frac{1}{2 a^{5} c^{3} x - 2 a^{4} c^{3}} + \frac{x}{a^{3} c^{3}} + \frac{\frac{5 \log{\left (x - \frac{1}{a} \right )}}{4} - \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a^{4} c^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15894, size = 69, normalized size = 1.21 \begin{align*} \frac{x}{c^{3}} - \frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac{5 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} - \frac{1}{2 \,{\left (a x - 1\right )} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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