Optimal. Leaf size=71 \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{1-x} x^2}{5 \sqrt{1-\frac{1}{x}}}-\frac{4 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{1-x} x}{15 \sqrt{1-\frac{1}{x}}} \]
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Rubi [A] time = 0.103072, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6176, 6181, 45, 37} \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{1-x} x^2}{5 \sqrt{1-\frac{1}{x}}}-\frac{4 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{1-x} x}{15 \sqrt{1-\frac{1}{x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} \sqrt{1-x} x \, dx &=\frac{\sqrt{1-x} \int e^{\coth ^{-1}(x)} \sqrt{1-\frac{1}{x}} x^{3/2} \, dx}{\sqrt{1-\frac{1}{x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{1-x} \sqrt{\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{x}}}\\ &=\frac{2 \left (1+\frac{1}{x}\right )^{3/2} \sqrt{1-x} x^2}{5 \sqrt{1-\frac{1}{x}}}+\frac{\left (2 \sqrt{1-x} \sqrt{\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 \sqrt{1-\frac{1}{x}}}\\ &=-\frac{4 \left (1+\frac{1}{x}\right )^{3/2} \sqrt{1-x} x}{15 \sqrt{1-\frac{1}{x}}}+\frac{2 \left (1+\frac{1}{x}\right )^{3/2} \sqrt{1-x} x^2}{5 \sqrt{1-\frac{1}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0150439, size = 41, normalized size = 0.58 \[ \frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) (3 x-2)}{15 \sqrt{\frac{x-1}{x}}} \]
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-2 \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 = 1.0517, size = 23, normalized size = 0.32 \begin{align*} \frac{1}{15} \,{\left (6 i \, x^{2} + 2 i \, x - 4 i\right )} \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57844, size = 99, normalized size = 1.39 \begin{align*} \frac{2 \,{\left (3 \, x^{3} + 4 \, x^{2} - x - 2\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 [C] time = 130.87, size = 46, normalized size = 0.65 \begin{align*} - \frac{14 i x}{15 \sqrt{\frac{1}{x + 1}}} - \frac{2 i \left (1 - x\right )^{2}}{5 \sqrt{\frac{1}{x + 1}}} + \frac{2 i}{3 \sqrt{\frac{1}{x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16351, size = 58, normalized size = 0.82 \begin{align*} -\frac{2}{15} \,{\left (2 i \, \sqrt{2} + \frac{3 \,{\left (x + 1\right )}^{2} \sqrt{-x - 1} + 5 \,{\left (-x - 1\right )}^{\frac{3}{2}}}{\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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