Integrand size = 12, antiderivative size = 107 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {28 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {86 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2} x}+\frac {2 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac {1}{x}\right )^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6311, 6316, 91, 79, 37} \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2 \sqrt {-\frac {1-x}{x}} x (x+1)^{3/2}}{5 \left (\frac {1}{x}+1\right )^{3/2}}+\frac {28 \sqrt {-\frac {1-x}{x}} (x+1)^{3/2}}{15 \left (\frac {1}{x}+1\right )^{3/2}}+\frac {86 \sqrt {-\frac {1-x}{x}} (x+1)^{3/2}}{15 \left (\frac {1}{x}+1\right )^{3/2} x} \]
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Rule 37
Rule 79
Rule 91
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{3/2} \int e^{\coth ^{-1}(x)} \left (1+\frac {1}{x}\right )^{3/2} x^{3/2} \, dx}{\left (1+\frac {1}{x}\right )^{3/2} x^{3/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {(1+x)^2}{\sqrt {1-x} x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {2 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac {1}{x}\right )^{3/2}}-\frac {\left (2 \left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {7+\frac {5 x}{2}}{\sqrt {1-x} x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 \left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {28 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {2 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac {1}{x}\right )^{3/2}}-\frac {\left (43 \left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{15 \left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {28 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {86 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2} x}+\frac {2 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac {1}{x}\right )^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.38 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} \left (43+14 x+3 x^2\right )}{15 \sqrt {1+\frac {1}{x}}} \]
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Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30
method | result | size |
gosper | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
default | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
risch | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.26 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2}{15} \, {\left (3 \, x^{2} + 14 \, x + 43\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
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Time = 42.55 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=2 \left (\begin {cases} 4 \sqrt {2} \left (\frac {\sqrt {2} \left (x - 1\right )^{\frac {5}{2}}}{40} + \frac {\sqrt {2} \left (x - 1\right )^{\frac {3}{2}}}{6} + \frac {\sqrt {2} \sqrt {x - 1}}{2}\right ) & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.21 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2 \, {\left (3 \, x^{3} + 11 \, x^{2} + 29 \, x - 43\right )}}{15 \, \sqrt {x - 1}} \]
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Exception generated. \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.36 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\sqrt {\frac {x-1}{x+1}}\,\left (\frac {28\,x\,\sqrt {x+1}}{15}+\frac {86\,\sqrt {x+1}}{15}+\frac {2\,x^2\,\sqrt {x+1}}{5}\right ) \]
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