Integrand size = 18, antiderivative size = 23 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6310, 6313, 272, 65, 214} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Rule 65
Rule 214
Rule 272
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 ) x} \, dx}{a c} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c} \\ & = -\frac {a \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c} \\ & = -\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\log \left (a x \left (1+\sqrt {\frac {-1+a^2 x^2}{a^2 x^2}}\right )\right )}{a c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(21)=42\).
Time = 0.43 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30
method | result | size |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c \sqrt {a^{2}}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x - 1}\, dx}{c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (21) = 42\).
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=-a {\left (\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c {\left | a \right |}} \]
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Time = 4.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \]
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