Optimal. Leaf size=32 \[ \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 32, 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} \[ \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \]
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^2} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{x^2} \, dx\\ &=\int \frac {(1+a x)^2}{x^2 (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {4 a}{x}+\frac {4 a^2}{(-1+a x)^2}-\frac {4 a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+\frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 1.00 \[ \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 55, normalized size = 1.72 \[ -\frac {5 \, a x + 4 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \, {\left (a^{2} x^{2} - a x\right )} \log \relax (x) - 1}{a x^{2} - x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 40, normalized size = 1.25 \[ 4 \, a \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right ) - \frac {4 \, a}{a x - 1} + \frac {a}{\frac {1}{a x - 1} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.97 \[ -\frac {1}{x}+4 a \ln \relax (x )-\frac {4 a}{a x -1}-4 a \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 1.06 \[ -4 \, a \log \left (a x - 1\right ) + 4 \, a \log \relax (x) - \frac {5 \, a x - 1}{a x^{2} - x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 28, normalized size = 0.88 \[ 8\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )+\frac {5\,a\,x-1}{x-a\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 26, normalized size = 0.81 \[ 4 a \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {- 5 a x + 1}{a x^{2} - x} \]
Verification of antiderivative is not currently implemented for this CAS.
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