Integrand size = 12, antiderivative size = 18 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{x}+2 a \log (x)-2 a \log (1+a x) \]
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Time = 0.03 (sec) , antiderivative size = 18, 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} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=2 a \log (x)-2 a \log (a x+1)+\frac {1}{x} \]
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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^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*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{x}+2 a \log (x)-2 a \log (1+a x) \]
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Time = 0.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {1}{x}+2 a \ln \left (x \right )-2 a \ln \left (a x +1\right )\) | \(19\) |
norman | \(\frac {1}{x}+2 a \ln \left (x \right )-2 a \ln \left (a x +1\right )\) | \(19\) |
risch | \(\frac {1}{x}+2 a \ln \left (-x \right )-2 a \ln \left (a x +1\right )\) | \(21\) |
parallelrisch | \(\frac {2 a \ln \left (x \right ) x -2 a \ln \left (a x +1\right ) x +1}{x}\) | \(23\) |
meijerg | \(a \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )\right )-a \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{a x}\right )\) | \(43\) |
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none
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {2 \, a x \log \left (a x + 1\right ) - 2 \, a x \log \left (x\right ) - 1}{x} \]
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Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=2 a \left (\log {\left (x \right )} - \log {\left (x + \frac {1}{a} \right )}\right ) + \frac {1}{x} \]
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none
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=-2 \, a \log \left (a x + 1\right ) + 2 \, a \log \left (x\right ) + \frac {1}{x} \]
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none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=-2 \, a \log \left ({\left | a x + 1 \right |}\right ) + 2 \, a \log \left ({\left | x \right |}\right ) + \frac {1}{x} \]
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Time = 4.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{x}-4\,a\,\mathrm {atanh}\left (2\,a\,x+1\right ) \]
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