Optimal. Leaf size=20 \[ \frac {2}{3} \sqrt {1-x} (x+1) e^{\coth ^{-1}(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.70 \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {1-x} x}{3 \sqrt {1-\frac {1}{x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 33, normalized size = 1.65 \[ \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.15, size = 27, normalized size = 1.35 \[ \frac {1}{3} \, {\left (-4 i \, \sqrt {2} + \frac {2 \, {\left (-x - 1\right )}^{\frac {3}{2}}}{\mathrm {sgn}\left (-x - 1\right )}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 1.20 \[ \frac {2 \left (1+x \right ) \sqrt {1-x}}{3 \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.50, size = 12, normalized size = 0.60 \[ \frac {1}{3} \, \sqrt {x + 1} {\left (2 i \, x + 2 i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 25, normalized size = 1.25 \[ -\frac {2\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2}{3\,\sqrt {1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.80, size = 29, normalized size = 1.45 \[ - \frac {2 i x}{3 \sqrt {\frac {1}{x + 1}}} - \frac {2 i}{3 \sqrt {\frac {1}{x + 1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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