Optimal. Leaf size=68 \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{14 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac{1}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.0991177, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6176, 6181, 78, 37} \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{5 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{14 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2}}{15 \left (1-\frac{1}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx &=\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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.0170106, size = 41, normalized size = 0.6 \[ -\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) (3 x-7)}{15 \sqrt{\frac{x-1}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 29, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2+2\,x \right ) \left ( 3\,x-7 \right ) }{15}\sqrt{1-x}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.05526, size = 23, normalized size = 0.34 \begin{align*} -\frac{1}{15} \,{\left (6 i \, x^{2} - 8 i \, x - 14 i\right )} \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62552, size = 101, normalized size = 1.49 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.17059, size = 59, normalized size = 0.87 \begin{align*} \frac{1}{15} \,{\left (-16 i \, \sqrt{2} + \frac{2 \,{\left (3 \,{\left (x + 1\right )}^{2} \sqrt{-x - 1} + 10 \,{\left (-x - 1\right )}^{\frac{3}{2}}\right )}}{\mathrm{sgn}\left (-x - 1\right )}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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