Optimal. Leaf size=104 \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{22 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{44 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac{1}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.126104, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6176, 6181, 78, 45, 37} \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac{1}{x}\right )^{3/2}}-\frac{22 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac{1}{x}\right )^{3/2}}+\frac{44 \left (\frac{1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac{1}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.0207203, size = 46, normalized size = 0.44 \[ -\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) \left (15 x^2-33 x+22\right )}{105 \sqrt{\frac{x-1}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 34, normalized size = 0.3 \begin{align*} -{\frac{ \left ( 2+2\,x \right ) \left ( 15\,{x}^{2}-33\,x+22 \right ) }{105}\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 = 1.06617, size = 30, normalized size = 0.29 \begin{align*} -\frac{1}{105} \,{\left (30 i \, x^{3} - 36 i \, x^{2} - 22 i \, x + 44 i\right )} \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62142, size = 120, normalized size = 1.15 \begin{align*} -\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 )}} \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 = 1.2415, size = 78, normalized size = 0.75 \begin{align*} \frac{1}{105} \,{\left (16 i \, \sqrt{2} + \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}}\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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