Optimal. Leaf size=51 \[ \frac{1}{4 a c^4 (1-a x)}+\frac{1}{4 a c^4 (1-a x)^2}+\frac{\tanh ^{-1}(a x)}{4 a c^4} \]
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Rubi [A] time = 0.0471335, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6129, 44, 207} \[ \frac{1}{4 a c^4 (1-a x)}+\frac{1}{4 a c^4 (1-a x)^2}+\frac{\tanh ^{-1}(a x)}{4 a c^4} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac{\int \frac{1}{(1-a x)^3 (1+a x)} \, dx}{c^4}\\ &=\frac{\int \left (-\frac{1}{2 (-1+a x)^3}+\frac{1}{4 (-1+a x)^2}-\frac{1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\frac{1}{4 a c^4 (1-a x)^2}+\frac{1}{4 a c^4 (1-a x)}-\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{4 c^4}\\ &=\frac{1}{4 a c^4 (1-a x)^2}+\frac{1}{4 a c^4 (1-a x)}+\frac{\tanh ^{-1}(a x)}{4 a c^4}\\ \end{align*}
Mathematica [A] time = 0.0224491, size = 35, normalized size = 0.69 \[ \frac{-a x+(a x-1)^2 \tanh ^{-1}(a x)+2}{4 a c^4 (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 60, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( ax+1 \right ) }{8\,a{c}^{4}}}+{\frac{1}{4\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{1}{4\,a{c}^{4} \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{8\,a{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956351, size = 85, normalized size = 1.67 \begin{align*} -\frac{a x - 2}{4 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac{\log \left (a x + 1\right )}{8 \, a c^{4}} - \frac{\log \left (a x - 1\right )}{8 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60785, size = 173, normalized size = 3.39 \begin{align*} -\frac{2 \, a x -{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) +{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 4}{8 \,{\left (a^{3} c^{4} x^{2} - 2 \, 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 = 0.499936, size = 56, normalized size = 1.1 \begin{align*} - \frac{a x - 2}{4 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 4 a c^{4}} - \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{8} - \frac{\log{\left (x + \frac{1}{a} \right )}}{8}}{a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20275, size = 78, normalized size = 1.53 \begin{align*} -\frac{\log \left ({\left | -\frac{2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{4}} - \frac{\frac{3}{a} - \frac{8}{{\left (a x + 1\right )} a}}{16 \, c^{4}{\left (\frac{2}{a x + 1} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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