Integrand size = 12, antiderivative size = 13 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=-\log (x)+2 \log (1+a x) \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6261, 78} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \log (a x+1)-\log (x) \]
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Rule 78
Rule 6261
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)}}{x} \, dx \\ & = -\int \frac {1-a x}{x (1+a x)} \, dx \\ & = -\int \left (\frac {1}{x}-\frac {2 a}{1+a x}\right ) \, dx \\ & = -\log (x)+2 \log (1+a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=-\log (x)+2 \log (1+a x) \]
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Time = 0.40 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\ln \left (x \right )+2 \ln \left (a x +1\right )\) | \(14\) |
norman | \(-\ln \left (x \right )+2 \ln \left (a x +1\right )\) | \(14\) |
parallelrisch | \(-\ln \left (x \right )+2 \ln \left (a x +1\right )\) | \(14\) |
risch | \(2 \ln \left (-a x -1\right )-\ln \left (x \right )\) | \(15\) |
meijerg | \(2 \ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )\) | \(18\) |
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none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \, \log \left (a x + 1\right ) - \log \left (x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=- \log {\left (x \right )} + 2 \log {\left (x + \frac {1}{a} \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \, \log \left (a x + 1\right ) - \log \left (x\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \, \log \left ({\left | a x + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2\,\ln \left (-3\,a\,x-3\right )-\ln \left (x\right ) \]
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