Integrand size = 15, antiderivative size = 104 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=\frac {44 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {22 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6311, 6316, 79, 47, 37} \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=\frac {2 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {22 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {44 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac {1}{x}\right )^{3/2}} \]
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Rule 37
Rule 47
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^{5/2} \, 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^{9/2}} \, 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^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {\left (11 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 \left (1-\frac {1}{x}\right )^{3/2}} \\ & = -\frac {22 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {\left (22 (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 )}{35 \left (1-\frac {1}{x}\right )^{3/2}} \\ & = \frac {44 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {22 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=-\frac {2 \sqrt {1+\frac {1}{x}} \sqrt {1-x} \left (22-11 x-18 x^2+15 x^3\right )}{105 \sqrt {\frac {-1+x}{x}}} \]
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Time = 0.44 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33
method | result | size |
gosper | \(-\frac {2 \left (1+x \right ) \sqrt {1-x}\, \left (15 x^{2}-33 x +22\right )}{105 \sqrt {\frac {x -1}{1+x}}}\) | \(34\) |
default | \(-\frac {2 \left (1+x \right ) \sqrt {1-x}\, \left (15 x^{2}-33 x +22\right )}{105 \sqrt {\frac {x -1}{1+x}}}\) | \(34\) |
risch | \(\frac {2 \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right ) \left (15 x^{3}-18 x^{2}-11 x +22\right )}{105 \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}\, \sqrt {-1-x}}\) | \(62\) |
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Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=-\frac {2 \, {\left (15 \, x^{4} - 3 \, x^{3} - 29 \, x^{2} + 11 \, x + 22\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{105 \, {\left (x - 1\right )}} \]
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Timed out. \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.21 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=\frac {2}{105} \, {\left (-15 i \, x^{3} + 18 i \, x^{2} + 11 i \, x - 22 i\right )} \sqrt {x + 1} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.61 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=\frac {16}{105} i \, \sqrt {2} \mathrm {sgn}\left (x + 1\right ) - \frac {2 \, {\left (15 \, {\left (x + 1\right )}^{3} \sqrt {-x - 1} - 63 \, {\left (x + 1\right )}^{2} \sqrt {-x - 1} - 70 \, {\left (-x - 1\right )}^{\frac {3}{2}} - 8 i \, \sqrt {2}\right )} \mathrm {sgn}\left (x\right )}{105 \, \mathrm {sgn}\left (x + 1\right )} \]
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Time = 4.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.34 \[ \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx=\frac {2\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2\,\left (15\,x^2-33\,x+22\right )}{105\,\sqrt {1-x}} \]
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