Integrand size = 18, antiderivative size = 48 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2}{a c (1-a x)^2}-\frac {4}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
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Time = 0.05 (sec) , antiderivative size = 48, 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} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {4}{a c (1-a x)}+\frac {2}{a c (1-a x)^2}-\frac {\log (1-a x)}{a c} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 \text {arctanh}(a x)}}{c-a c x} \, dx \\ & = \frac {\int \frac {(1+a x)^2}{(1-a x)^3} \, dx}{c} \\ & = \frac {\int \left (\frac {1}{1-a x}-\frac {4}{(-1+a x)^3}-\frac {4}{(-1+a x)^2}\right ) \, dx}{c} \\ & = \frac {2}{a c (1-a x)^2}-\frac {4}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {-2+4 a x-(-1+a x)^2 \log (1-a x)}{a c (-1+a x)^2} \]
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Time = 0.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {2 a \,x^{2}}{c \left (a x -1\right )^{2}}-\frac {\ln \left (a x -1\right )}{a c}\) | \(32\) |
risch | \(\frac {4 x -\frac {2}{a}}{c \left (a x -1\right )^{2}}-\frac {\ln \left (a x -1\right )}{a c}\) | \(36\) |
default | \(\frac {\frac {2}{\left (a x -1\right )^{2} a}+\frac {4}{a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a}}{c}\) | \(41\) |
parallelrisch | \(\frac {-a^{2} \ln \left (a x -1\right ) x^{2}+2 a^{2} x^{2}+2 a \ln \left (a x -1\right ) x -\ln \left (a x -1\right )}{\left (a x -1\right )^{2} c a}\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {4 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 2}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} \]
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Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {- 4 a x + 2}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac {\log {\left (a x - 1 \right )}}{a c} \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 \, {\left (2 \, a x - 1\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} - \frac {\log \left (a x - 1\right )}{a c} \]
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Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c} + \frac {2 \, {\left (\frac {2 \, a c}{a x - 1} + \frac {a c}{{\left (a x - 1\right )}^{2}}\right )}}{a^{2} c^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {4\,x-\frac {2}{a}}{c\,a^2\,x^2-2\,c\,a\,x+c}-\frac {\ln \left (a\,x-1\right )}{a\,c} \]
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