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 = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 88
Rule 6126
Rule 6167
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*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.00 \[ \frac {4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac {4 a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 73, normalized size = 1.59 \[ -\frac {16 \, a^{2} x^{2} - 7 \, a x + 16 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 16 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \relax (x) - 1}{2 \, {\left (a x^{3} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 62, normalized size = 1.35 \[ 8 \, a^{2} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right ) - \frac {4 \, a^{2}}{a x - 1} + \frac {9 \, a^{2} + \frac {10 \, a^{2}}{a x - 1}}{2 \, {\left (\frac {1}{a x - 1} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.93 \[ -\frac {1}{2 x^{2}}-\frac {4 a}{x}+8 a^{2} \ln \relax (x )-\frac {4 a^{2}}{a x -1}-8 a^{2} \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 48, normalized size = 1.04 \[ -8 \, a^{2} \log \left (a x - 1\right ) + 8 \, a^{2} \log \relax (x) - \frac {16 \, a^{2} x^{2} - 7 \, a x - 1}{2 \, {\left (a x^{3} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 41, normalized size = 0.89 \[ 16\,a^2\,\mathrm {atanh}\left (2\,a\,x-1\right )+\frac {-8\,a^2\,x^2+\frac {7\,a\,x}{2}+\frac {1}{2}}{a\,x^3-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 41, normalized size = 0.89 \[ 8 a^{2} \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {- 16 a^{2} x^{2} + 7 a x + 1}{2 a x^{3} - 2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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