Optimal. Leaf size=20 \[ -2 a \log (x)+2 a \log (a x+1)-\frac {1}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 20, 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} \[ -2 a \log (x)+2 a \log (a x+1)-\frac {1}{x} \]
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 = 20, normalized size = 1.00 \[ -2 a \log (x)+2 a \log (a x+1)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 22, 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 = 0.17, size = 30, normalized size = 1.50 \[ -2 \, a \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right ) + \frac {a}{\frac {1}{a x + 1} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 21, normalized size = 1.05 \[ -\frac {1}{x}-2 a \ln \relax (x )+2 a \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 1.00 \[ 2 \, a \log \left (a x + 1\right ) - 2 \, a \log \relax (x) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 16, normalized size = 0.80 \[ 4\,a\,\mathrm {atanh}\left (2\,a\,x+1\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 17, normalized size = 0.85 \[ - 2 a \left (\log {\relax (x )} - \log {\left (x + \frac {1}{a} \right )}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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