Integrand size = 12, antiderivative size = 53 \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\frac {1}{2} \text {arctanh}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6310, 6313, 821, 272, 43, 65, 212} \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=\frac {1}{2} \text {arctanh}\left (\sqrt {1-\frac {1}{x^2}}\right )-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3 \]
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Rule 43
Rule 65
Rule 212
Rule 272
Rule 821
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \int e^{\coth ^{-1}(x)} \left (1-\frac {1}{x}\right )^2 x^2 \, dx \\ & = -\text {Subst}\left (\int \frac {(1-x) \sqrt {1-x^2}}{x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x}}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x^2}}\right ) \\ & = -\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\frac {1}{2} \text {arctanh}\left (\sqrt {1-\frac {1}{x^2}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x \left (-2-3 x+2 x^2\right )+\frac {1}{2} \log \left (\left (1+\sqrt {1-\frac {1}{x^2}}\right ) x\right ) \]
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Time = 0.48 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\left (x -1\right ) \left (2 \left (\left (x -1\right ) \left (1+x \right )\right )^{\frac {3}{2}}-3 x \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{6 \sqrt {\frac {x -1}{1+x}}\, \sqrt {\left (x -1\right ) \left (1+x \right )}}\) | \(60\) |
risch | \(\frac {\left (2 x^{2}-3 x -2\right ) \left (x -1\right )}{6 \sqrt {\frac {x -1}{1+x}}}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (x -1\right ) \left (1+x \right )}}{2 \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right )}\) | \(65\) |
trager | \(\frac {\left (1+x \right ) \left (2 x^{2}-3 x -2\right ) \sqrt {-\frac {1-x}{1+x}}}{6}-\frac {\ln \left (-\sqrt {-\frac {1-x}{1+x}}\, x -\sqrt {-\frac {1-x}{1+x}}+x \right )}{2}\) | \(69\) |
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Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.15 \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=\frac {1}{6} \, {\left (2 \, x^{3} - x^{2} - 5 \, x - 2\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
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\[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=\int \frac {\left (x - 1\right )^{2}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (41) = 82\).
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.11 \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=-\frac {3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{2}} + 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} - 3 \, \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.13 \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=\frac {1}{6} \, \sqrt {x^{2} - 1} {\left (x {\left (\frac {2 \, x}{\mathrm {sgn}\left (x + 1\right )} - \frac {3}{\mathrm {sgn}\left (x + 1\right )}\right )} - \frac {2}{\mathrm {sgn}\left (x + 1\right )}\right )} - \frac {\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{2 \, \mathrm {sgn}\left (x + 1\right )} \]
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Time = 4.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.70 \[ \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx=\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{3}-\sqrt {\frac {x-1}{x+1}}+{\left (\frac {x-1}{x+1}\right )}^{5/2}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1} \]
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