Integrand size = 18, antiderivative size = 28 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \left (a-\frac {1}{x}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 28, 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 = {6310, 6313, 665} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \left (a-\frac {1}{x}\right )} \]
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Rule 665
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^2 x^2} \, dx}{a^2 c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2 c^2} \\ & = -\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \left (a-\frac {1}{x}\right )} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 (-1+a x)} \]
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Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29
method | result | size |
gosper | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}}\) | \(36\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}}\) | \(36\) |
trager | \(-\frac {\left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{a \,c^{2} \left (a x -1\right )}\) | \(38\) |
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {1}{a c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a c x - c\right )}^{2}} \,d x } \]
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Time = 4.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {1}{a\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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