Integrand size = 13, antiderivative size = 55 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\text {arctanh}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {6310, 6313, 866, 1819, 837, 12, 272, 65, 212} \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=\text {arctanh}\left (\sqrt {1-\frac {1}{x^2}}\right )-\frac {4 \left (\frac {1}{x}+1\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {\frac {5}{x}+3}{3 \sqrt {1-\frac {1}{x^2}}} \]
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Rule 12
Rule 65
Rule 212
Rule 272
Rule 837
Rule 866
Rule 1819
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\coth ^{-1}(x)}}{\left (1-\frac {1}{x}\right )^2 x} \, dx \\ & = -\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{(1-x)^3 x} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \frac {(1+x)^3}{x \left (1-x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \text {Subst}\left (\int \frac {-3-5 x}{x \left (1-x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\frac {1}{3} \text {Subst}\left (\int -\frac {3}{x \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}-\text {Subst}\left (\int \frac {1}{x \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x^2}}\right ) \\ & = -\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\text {arctanh}\left (\sqrt {1-\frac {1}{x^2}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=\frac {\sqrt {1-\frac {1}{x^2}} (5-7 x) x}{3 (-1+x)^2}+\log \left (\left (1+\sqrt {1-\frac {1}{x^2}}\right ) x\right ) \]
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Time = 0.48 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16
method | result | size |
trager | \(-\frac {\left (1+x \right ) \left (7 x -5\right ) \sqrt {-\frac {1-x}{1+x}}}{3 \left (x -1\right )^{2}}+\ln \left (\sqrt {-\frac {1-x}{1+x}}\, x +\sqrt {-\frac {1-x}{1+x}}+x \right )\) | \(64\) |
risch | \(-\frac {7 x^{2}+2 x -5}{3 \left (x -1\right ) \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right )}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (x -1\right ) \left (1+x \right )}}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right )}\) | \(71\) |
default | \(-\frac {3 x \left (x^{2}-1\right )^{\frac {3}{2}}-3 \sqrt {x^{2}-1}\, x^{3}-3 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{3}-2 \left (x^{2}-1\right )^{\frac {3}{2}}+9 x^{2} \sqrt {x^{2}-1}+9 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{2}-9 x \sqrt {x^{2}-1}-9 \ln \left (x +\sqrt {x^{2}-1}\right ) x +3 \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )}{3 \left (x -1\right )^{2} \sqrt {\left (x -1\right ) \left (1+x \right )}\, \sqrt {\frac {x -1}{1+x}}}\) | \(146\) |
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Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=\frac {3 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - 3 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) - {\left (7 \, x^{2} + 2 \, x - 5\right )} \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=\int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \left (x - 1\right )^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=-\frac {\frac {6 \, {\left (x - 1\right )}}{x + 1} + 1}{3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=-\frac {\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {2 \, {\left (9 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 12 \, x + 12 \, \sqrt {x^{2} - 1} + 7\right )}}{3 \, {\left (x - \sqrt {x^{2} - 1} - 1\right )}^{3} \mathrm {sgn}\left (x + 1\right )} + \frac {7}{3} \, \mathrm {sgn}\left (x + 1\right ) \]
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Time = 4.62 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx=2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {\frac {2\,\left (x-1\right )}{x+1}+\frac {1}{3}}{{\left (\frac {x-1}{x+1}\right )}^{3/2}} \]
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