Integrand size = 14, antiderivative size = 68 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {14 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6311, 6316, 79, 37} \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\frac {2 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {14 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac {1}{x}\right )^{3/2}} \]
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Rule 37
Rule 79
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 ((1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {(1-x) \sqrt {1+x}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{x}\right )^{3/2}} \\ & = \frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {\left (7 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 \left (1-\frac {1}{x}\right )^{3/2}} \\ & = -\frac {14 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac {1}{x}\right )^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.63 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {2 \sqrt {1+\frac {1}{x}} \sqrt {1-x} \left (-7-4 x+3 x^2\right )}{15 \sqrt {\frac {-1+x}{x}}} \]
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Time = 0.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 \left (1+x \right ) \left (3 x -7\right ) \sqrt {1-x}}{15 \sqrt {\frac {x -1}{1+x}}}\) | \(29\) |
default | \(-\frac {2 \left (1+x \right ) \left (3 x -7\right ) \sqrt {1-x}}{15 \sqrt {\frac {x -1}{1+x}}}\) | \(29\) |
risch | \(\frac {2 \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right ) \left (3 x^{2}-4 x -7\right )}{15 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}\, \sqrt {-1-x}}\) | \(57\) |
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none
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, x^{3} - x^{2} - 11 \, x - 7\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{15 \, {\left (x - 1\right )}} \]
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\[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\int \frac {\left (1 - x\right )^{\frac {3}{2}}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.25 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\frac {2}{15} \, {\left (-3 i \, x^{2} + 4 i \, x + 7 i\right )} \sqrt {x + 1} \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=-\frac {16}{15} i \, \sqrt {2} \mathrm {sgn}\left (x + 1\right ) - \frac {2 \, {\left (3 \, {\left (x + 1\right )}^{2} \sqrt {-x - 1} + 10 \, {\left (-x - 1\right )}^{\frac {3}{2}} + 8 i \, \sqrt {2}\right )} \mathrm {sgn}\left (x\right )}{15 \, \mathrm {sgn}\left (x + 1\right )} \]
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Time = 4.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.44 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx=\frac {2\,\left (3\,x-7\right )\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2}{15\,\sqrt {1-x}} \]
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