Optimal. Leaf size=32 \[ -2 a^2 \log (x)+2 a^2 \log (a x+1)-\frac {2 a}{x}+\frac {1}{2 x^2} \]
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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, 77} \[ -2 a^2 \log (x)+2 a^2 \log (a x+1)-\frac {2 a}{x}+\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^3} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{x^3} \, dx\\ &=-\int \frac {1-a x}{x^3 (1+a x)} \, dx\\ &=-\int \left (\frac {1}{x^3}-\frac {2 a}{x^2}+\frac {2 a^2}{x}-\frac {2 a^3}{1+a x}\right ) \, dx\\ &=\frac {1}{2 x^2}-\frac {2 a}{x}-2 a^2 \log (x)+2 a^2 \log (1+a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ -2 a^2 \log (x)+2 a^2 \log (a x+1)-\frac {2 a}{x}+\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 35, normalized size = 1.09 \[ \frac {4 \, a^{2} x^{2} \log \left (a x + 1\right ) - 4 \, a^{2} x^{2} \log \relax (x) - 4 \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 1.00 \[ 2 \, a^{2} \log \left ({\left | a x + 1 \right |}\right ) - 2 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a x - 1}{2 \, x^{2}} \]
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 \[ \frac {1}{2 x^{2}}-\frac {2 a}{x}-2 a^{2} \ln \relax (x )+2 a^{2} \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 0.94 \[ 2 \, a^{2} \log \left (a x + 1\right ) - 2 \, a^{2} \log \relax (x) - \frac {4 \, a x - 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.75 \[ 4\,a^2\,\mathrm {atanh}\left (2\,a\,x+1\right )-\frac {2\,a\,x-\frac {1}{2}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 26, normalized size = 0.81 \[ 2 a^{2} \left (- \log {\relax (x )} + \log {\left (x + \frac {1}{a} \right )}\right ) + \frac {- 4 a x + 1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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