Optimal. Leaf size=20 \[ \frac{2}{3} \sqrt{1-x} (x+1) e^{\coth ^{-1}(x)} \]
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Rubi [A] time = 0.0196056, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {6174} \[ \frac{2}{3} \sqrt{1-x} (x+1) e^{\coth ^{-1}(x)} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} \sqrt{1-x} \, dx &=\frac{2}{3} e^{\coth ^{-1}(x)} \sqrt{1-x} (1+x)\\ \end{align*}
Mathematica [A] time = 0.0124854, size = 34, normalized size = 1.7 \[ \frac{2 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{1-x} x}{3 \sqrt{1-\frac{1}{x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.056, size = 24, normalized size = 1.2 \begin{align*}{\frac{2+2\,x}{3}\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 = 1.09669, size = 16, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \sqrt{x + 1}{\left (2 i \, x + 2 i\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59786, size = 86, normalized size = 4.3 \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{3 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 48.8617, size = 29, normalized size = 1.45 \begin{align*} - \frac{2 i x}{3 \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 = 1.14028, size = 36, normalized size = 1.8 \begin{align*} -\frac{2}{3} \,{\left (2 i \, \sqrt{2} - \frac{{\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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