Integrand size = 8, antiderivative size = 13 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x-\frac {2 \log (1+a x)}{a} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6260, 45} \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x-\frac {2 \log (a x+1)}{a} \]
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Rule 45
Rule 6260
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \, dx \\ & = -\int \frac {1-a x}{1+a x} \, dx \\ & = -\int \left (-1+\frac {2}{1+a x}\right ) \, dx \\ & = x-\frac {2 \log (1+a x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x-\frac {2 \log (1+a x)}{a} \]
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Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
default | \(x -\frac {2 \ln \left (a x +1\right )}{a}\) | \(14\) |
norman | \(x -\frac {2 \ln \left (a x +1\right )}{a}\) | \(14\) |
risch | \(x -\frac {2 \ln \left (a x +1\right )}{a}\) | \(14\) |
parallelrisch | \(-\frac {-a x +2 \ln \left (a x +1\right )}{a}\) | \(19\) |
meijerg | \(\frac {a x -\ln \left (a x +1\right )}{a}-\frac {\ln \left (a x +1\right )}{a}\) | \(29\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=\frac {a x - 2 \, \log \left (a x + 1\right )}{a} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x - \frac {2 \log {\left (a x + 1 \right )}}{a} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x - \frac {2 \, \log \left (a x + 1\right )}{a} \]
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none
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x - \frac {2 \, \log \left ({\left | a x + 1 \right |}\right )}{a} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \, dx=x-\frac {2\,\ln \left (a\,x+1\right )}{a} \]
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