Integrand size = 22, antiderivative size = 16 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {1}{a c (1-a x)} \]
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Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 32} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {1}{a c (1-a x)} \]
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Rule 32
Rule 6275
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{c-a^2 c x^2} \, dx \\ & = -\frac {\int \frac {1}{(1-a x)^2} \, dx}{c} \\ & = -\frac {1}{a c (1-a x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{2 \coth ^{-1}(a x)}}{2 a c} \]
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Time = 0.57 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {x}{c \left (a x -1\right )}\) | \(13\) |
parallelrisch | \(\frac {x}{c \left (a x -1\right )}\) | \(13\) |
gosper | \(\frac {1}{a c \left (a x -1\right )}\) | \(15\) |
default | \(\frac {1}{a c \left (a x -1\right )}\) | \(15\) |
risch | \(\frac {1}{a c \left (a x -1\right )}\) | \(15\) |
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{a^{2} c x - a c} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{a^{2} c x - a c} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{a^{2} c x - a c} \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{{\left (a x - 1\right )} a c} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {1}{a\,\left (c-a\,c\,x\right )} \]
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