Integrand size = 18, antiderivative size = 32 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 45} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{c-a c x} \, dx \\ & = -\frac {\int \frac {1+a x}{(1-a x)^2} \, dx}{c} \\ & = -\frac {\int \left (\frac {2}{(-1+a x)^2}+\frac {1}{-1+a x}\right ) \, dx}{c} \\ & = -\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\frac {2}{a (1-a x)}+\frac {\log (1-a x)}{a}}{c} \]
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Time = 0.52 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\frac {2}{a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a}}{c}\) | \(29\) |
norman | \(\frac {2 x}{c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a c}\) | \(29\) |
risch | \(\frac {2}{a c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a c}\) | \(31\) |
parallelrisch | \(\frac {-a \ln \left (a x -1\right ) x +2 a x +\ln \left (a x -1\right )}{c \left (a x -1\right ) a}\) | \(36\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2}{a^{2} c x - a c} - \frac {\log {\left (a x - 1 \right )}}{a c} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2}{a^{2} c x - a c} - \frac {\log \left (a x - 1\right )}{a c} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac {2}{{\left (a x - 1\right )} a c} \]
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Time = 4.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {2}{a\,\left (c-a\,c\,x\right )}-\frac {\ln \left (a\,x-1\right )}{a\,c} \]
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