Integrand size = 12, antiderivative size = 24 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{3 \left (1-\frac {1}{x}\right )^3} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6310, 6313, 665} \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{3 \left (1-\frac {1}{x}\right )^3} \]
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Rule 665
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\coth ^{-1}(x)}}{\left (1-\frac {1}{x}\right )^2 x^2} \, dx \\ & = -\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{3 \left (1-\frac {1}{x}\right )^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {\sqrt {1-\frac {1}{x^2}} x (1+x)}{3 (-1+x)^2} \]
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Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(-\frac {1+x}{3 \left (x -1\right ) \sqrt {\frac {x -1}{1+x}}}\) | \(22\) |
trager | \(-\frac {\left (1+x \right )^{2} \sqrt {-\frac {1-x}{1+x}}}{3 \left (x -1\right )^{2}}\) | \(27\) |
risch | \(-\frac {x^{2}+2 x +1}{3 \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \left (x -1\right )}\) | \(32\) |
default | \(-\frac {\left (x^{2}-1\right )^{\frac {3}{2}}}{3 \sqrt {\frac {x -1}{1+x}}\, \left (x -1\right )^{2} \sqrt {\left (x -1\right ) \left (1+x \right )}}\) | \(35\) |
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {{\left (x^{2} + 2 \, x + 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=\int \frac {1}{\sqrt {\frac {x - 1}{x + 1}} \left (x - 1\right )^{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {1}{3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=\frac {2 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}}{3 \, {\left (x - \sqrt {x^{2} - 1} - 1\right )}^{3} \mathrm {sgn}\left (x + 1\right )} + \frac {1}{3} \, \mathrm {sgn}\left (x + 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {1}{3\,{\left (\frac {x-1}{x+1}\right )}^{3/2}} \]
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