Optimal. Leaf size=58 \[ -\frac{1}{2 a c^3 (1-a x)}-\frac{5 \log (1-a x)}{4 a c^3}+\frac{\log (a x+1)}{4 a c^3}-\frac{x}{c^3} \]
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Rubi [A] time = 0.124837, antiderivative size = 58, 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, 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 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \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 = 0.107489, size = 57, 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 = 0.038, size = 51, 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 = 0.956659, size = 73, 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 = 2.21859, size = 138, normalized size = 2.38 \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 = 0.556586, size = 58, normalized size = 1. \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 = 1.23157, size = 115, normalized size = 1.98 \begin{align*} -\frac{{\left (a x + 1\right )}{\left (\frac{9}{a x + 1} - 4\right )}}{4 \, a c^{3}{\left (\frac{2}{a x + 1} - 1\right )}} + \frac{\log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a c^{3}} - \frac{5 \, \log \left ({\left | -\frac{2}{a x + 1} + 1 \right |}\right )}{4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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