Integrand size = 12, antiderivative size = 70 \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\frac {10 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}} \]
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Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6311, 6316, 79, 37} \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{3 \sqrt {\frac {1}{x}+1}}+\frac {10 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{3 \sqrt {\frac {1}{x}+1}} \]
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Rule 37
Rule 79
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x} \int e^{\coth ^{-1}(x)} \sqrt {1+\frac {1}{x}} \sqrt {x} \, dx}{\sqrt {1+\frac {1}{x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1+x}{\sqrt {1-x} x^{5/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1+\frac {1}{x}}} \\ & = \frac {2 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}}-\frac {\left (5 \sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 \sqrt {1+\frac {1}{x}}} \\ & = \frac {10 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.49 \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} (5+x)}{3 \sqrt {1+\frac {1}{x}}} \]
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Time = 0.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(\frac {2 \left (x -1\right ) \left (x +5\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(25\) |
default | \(\frac {2 \left (x -1\right ) \left (x +5\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(25\) |
risch | \(\frac {2 \left (x -1\right ) \left (x +5\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1+x}}\) | \(25\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.30 \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\frac {2}{3} \, {\left (x + 5\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
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Time = 2.95 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.80 \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=2 \left (\begin {cases} 2 \sqrt {2} \left (\frac {\sqrt {2} \left (x - 1\right )^{\frac {3}{2}}}{12} + \frac {\sqrt {2} \sqrt {x - 1}}{2}\right ) & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.21 \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\frac {2 \, {\left (x^{2} + 4 \, x - 5\right )}}{3 \, \sqrt {x - 1}} \]
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Exception generated. \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.30 \[ \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx=\frac {2\,\sqrt {\frac {x-1}{x+1}}\,\sqrt {x+1}\,\left (x+5\right )}{3} \]
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