Integrand size = 12, antiderivative size = 33 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\frac {2 \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x}{\sqrt {1+x}} \]
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Time = 0.07 (sec) , antiderivative size = 33, 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 = {6311, 6316, 37} \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\frac {2 \sqrt {\frac {1}{x}+1} \sqrt {-\frac {1-x}{x}} x}{\sqrt {x+1}} \]
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Rule 37
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 {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {1+x}} \\ & = \frac {2 \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x}{\sqrt {1+x}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x^2}} x}{\sqrt {1+x}} \]
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Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(22\) |
default | \(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(22\) |
risch | \(\frac {2 x -2}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(22\) |
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none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=2 \, \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
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Result contains complex when optimal does not.
Time = 6.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=\begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.21 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=2 \, \sqrt {x - 1} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=-\frac {2 \, {\left (i \, \sqrt {2} - \sqrt {x - 1}\right )} \mathrm {sgn}\left (x\right )}{\mathrm {sgn}\left (x + 1\right )} \]
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Time = 4.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1+x}} \, dx=2\,\sqrt {\frac {x-1}{x+1}}\,\sqrt {x+1} \]
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