Integrand size = 14, antiderivative size = 90 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]
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Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]
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Rule 95
Rule 96
Rule 212
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-\frac {1}{x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-\frac {1}{x}} \sqrt {x}} \, dx}{\sqrt {1-x}} \\ & = -\frac {\sqrt {1-\frac {1}{x}} \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \\ & = \frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}-\frac {\left (2 \sqrt {1-\frac {1}{x}}\right ) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \\ & = \frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}-\frac {\left (4 \sqrt {1-\frac {1}{x}}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \\ & = \frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} x \left (\sqrt {1+\frac {1}{x}}-\sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{\sqrt {1-x}} \]
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Time = 0.46 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {2 \sqrt {1-x}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-\sqrt {-1-x}\right )}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {-1-x}}\) | \(55\) |
risch | \(\frac {2 \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {-1-x}\, \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\) | \(108\) |
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Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{x - 1} \]
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Time = 13.99 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=- 2 \left (\begin {cases} \sqrt {2} \left (\frac {\sqrt {2} \sqrt {- x - 1}}{2} - \operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}\right ) & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \]
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\[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\int { \frac {1}{\sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\text {Exception raised: NotImplementedError} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\int \frac {1}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \]
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