Optimal. Leaf size=70 \[ -\frac{5}{4 a^4 c^2 (1-a x)}+\frac{1}{4 a^4 c^2 (1-a x)^2}-\frac{7 \log (1-a x)}{8 a^4 c^2}-\frac{\log (a x+1)}{8 a^4 c^2} \]
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Rubi [A] time = 0.106701, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 88} \[ -\frac{5}{4 a^4 c^2 (1-a x)}+\frac{1}{4 a^4 c^2 (1-a x)^2}-\frac{7 \log (1-a x)}{8 a^4 c^2}-\frac{\log (a x+1)}{8 a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{x^3}{(1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac{\int \left (-\frac{1}{2 a^3 (-1+a x)^3}-\frac{5}{4 a^3 (-1+a x)^2}-\frac{7}{8 a^3 (-1+a x)}-\frac{1}{8 a^3 (1+a x)}\right ) \, dx}{c^2}\\ &=\frac{1}{4 a^4 c^2 (1-a x)^2}-\frac{5}{4 a^4 c^2 (1-a x)}-\frac{7 \log (1-a x)}{8 a^4 c^2}-\frac{\log (1+a x)}{8 a^4 c^2}\\ \end{align*}
Mathematica [A] time = 0.0358648, size = 53, normalized size = 0.76 \[ -\frac{-10 a x+7 (a x-1)^2 \log (1-a x)+(a x-1)^2 \log (a x+1)+8}{8 a^4 c^2 (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 60, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{8\,{a}^{4}{c}^{2}}}+{\frac{1}{4\,{a}^{4}{c}^{2} \left ( ax-1 \right ) ^{2}}}+{\frac{5}{4\,{a}^{4}{c}^{2} \left ( ax-1 \right ) }}-{\frac{7\,\ln \left ( ax-1 \right ) }{8\,{a}^{4}{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958823, size = 89, normalized size = 1.27 \begin{align*} \frac{5 \, a x - 4}{4 \,{\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}} - \frac{\log \left (a x + 1\right )}{8 \, a^{4} c^{2}} - \frac{7 \, \log \left (a x - 1\right )}{8 \, a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22161, size = 178, normalized size = 2.54 \begin{align*} \frac{10 \, a x -{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) - 7 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 8}{8 \,{\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.537136, size = 61, normalized size = 0.87 \begin{align*} \frac{5 a x - 4}{4 a^{6} c^{2} x^{2} - 8 a^{5} c^{2} x + 4 a^{4} c^{2}} - \frac{\frac{7 \log{\left (x - \frac{1}{a} \right )}}{8} + \frac{\log{\left (x + \frac{1}{a} \right )}}{8}}{a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.141, size = 70, normalized size = 1. \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{4} c^{2}} - \frac{7 \, \log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{4} c^{2}} + \frac{5 \, a x - 4}{4 \,{\left (a x - 1\right )}^{2} a^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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