Optimal. Leaf size=46 \[ \frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.054295, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6126, 88} \[ \frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{x^3} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{x^3} \, dx\\ &=\int \frac{(1+a x)^2}{x^3 (1-a x)^2} \, dx\\ &=\int \left (\frac{1}{x^3}+\frac{4 a}{x^2}+\frac{8 a^2}{x}+\frac{4 a^3}{(-1+a x)^2}-\frac{8 a^3}{-1+a x}\right ) \, dx\\ &=-\frac{1}{2 x^2}-\frac{4 a}{x}+\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.0300503, size = 46, normalized size = 1. \[ \frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 43, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-4\,{\frac{a}{x}}+8\,{a}^{2}\ln \left ( x \right ) -4\,{\frac{{a}^{2}}{ax-1}}-8\,{a}^{2}\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01859, size = 65, normalized size = 1.41 \begin{align*} -8 \, a^{2} \log \left (a x - 1\right ) + 8 \, a^{2} \log \left (x\right ) - \frac{16 \, a^{2} x^{2} - 7 \, a x - 1}{2 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81316, size = 155, normalized size = 3.37 \begin{align*} -\frac{16 \, a^{2} x^{2} - 7 \, a x + 16 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 16 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 1}{2 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.459218, size = 41, normalized size = 0.89 \begin{align*} 8 a^{2} \left (\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{16 a^{2} x^{2} - 7 a x - 1}{2 a x^{3} - 2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15712, size = 84, normalized size = 1.83 \begin{align*} 8 \, a^{2} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right ) - \frac{4 \, a^{2}}{a x - 1} + \frac{9 \, a^{2} + \frac{10 \, a^{2}}{a x - 1}}{2 \,{\left (\frac{1}{a x - 1} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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