Integrand size = 8, antiderivative size = 14 \[ \int e^{2 \coth ^{-1}(a x)} \, dx=x+\frac {2 \log (1-a x)}{a} \]
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Time = 0.02 (sec) , antiderivative size = 14, 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} \[ \int e^{2 \coth ^{-1}(a x)} \, dx=\frac {2 \log (1-a x)}{a}+x \]
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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 = 14, 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 = 0.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
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 | \(\frac {a x +2 \ln \left (a x -1\right )}{a}\) | \(17\) |
meijerg | \(\frac {\ln \left (-a x +1\right )}{a}-\frac {-a x -\ln \left (-a x +1\right )}{a}\) | \(32\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int e^{2 \coth ^{-1}(a x)} \, dx=\frac {a x + 2 \, \log \left (a x - 1\right )}{a} \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int e^{2 \coth ^{-1}(a x)} \, dx=x + \frac {2 \log {\left (a x - 1 \right )}}{a} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int e^{2 \coth ^{-1}(a x)} \, dx=x + \frac {2 \, \log \left (a x - 1\right )}{a} \]
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none
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \, dx=x + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a} \]
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Time = 4.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int e^{2 \coth ^{-1}(a x)} \, dx=x+\frac {2\,\ln \left (a\,x-1\right )}{a} \]
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