Optimal. Leaf size=21 \[ 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6126, 77} \[ 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {1+a x}{x^2 (1-a x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {2 a}{x}-\frac {2 a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+2 a \log (x)-2 a \log (1-a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 23, normalized size = 1.10 \[ -\frac {2 \, a x \log \left (a x - 1\right ) - 2 \, a x \log \relax (x) + 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 22, normalized size = 1.05 \[ -2 \, a \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a \log \left ({\left | x \right |}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 21, normalized size = 1.00 \[ -\frac {1}{x}+2 a \ln \relax (x )-2 a \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 0.95 \[ -2 \, a \log \left (a x - 1\right ) + 2 \, a \log \relax (x) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 16, normalized size = 0.76 \[ 4\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 17, normalized size = 0.81 \[ - 2 a \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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