Integrand size = 18, antiderivative size = 14 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {x}{c^2 (1-a x)^2} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 34} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {x}{c^2 (1-a x)^2} \]
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Rule 34
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{(c-a c x)^2} \, dx \\ & = -\frac {\int \frac {1+a x}{(1-a x)^3} \, dx}{c^2} \\ & = -\frac {x}{c^2 (1-a x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {(1+a x)^2}{4 a c^2 (1-a x)^2} \]
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Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(-\frac {x}{c^{2} \left (a x -1\right )^{2}}\) | \(14\) |
norman | \(-\frac {x}{c^{2} \left (a x -1\right )^{2}}\) | \(14\) |
risch | \(-\frac {x}{c^{2} \left (a x -1\right )^{2}}\) | \(14\) |
parallelrisch | \(-\frac {x}{c^{2} \left (a x -1\right )^{2}}\) | \(14\) |
default | \(\frac {-\frac {1}{\left (a x -1\right )^{2} a}-\frac {1}{a \left (a x -1\right )}}{c^{2}}\) | \(30\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=- \frac {x}{a^{2} c^{2} x^{2} - 2 a c^{2} x + c^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {1}{{\left (a c x - c\right )}^{2} a} - \frac {1}{{\left (a c x - c\right )} a c} \]
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Time = 4.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {x}{c^2\,{\left (a\,x-1\right )}^2} \]
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