Optimal. Leaf size=90 \[ \frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}} \]
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Rubi [A] time = 0.0925382, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6176, 6181, 94, 93, 206} \[ \frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)}}{\sqrt{1-x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{x}} \sqrt{x}\right ) \int \frac{e^{\coth ^{-1}(x)}}{\sqrt{1-\frac{1}{x}} \sqrt{x}} \, dx}{\sqrt{1-x}}\\ &=-\frac{\sqrt{1-\frac{1}{x}} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{(1-x) x^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}-\frac{\left (2 \sqrt{1-\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{x} \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}-\frac{\left (4 \sqrt{1-\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0289919, size = 63, normalized size = 0.7 \[ \frac{2 \sqrt{\frac{x-1}{x}} x \left (\sqrt{\frac{1}{x}+1}-\sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{x+1}}\right )\right )}{\sqrt{1-x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.121, size = 55, normalized size = 0.6 \begin{align*} 2\,{\frac{\sqrt{1-x} \left ( \sqrt{2}\arctan \left ( 1/2\,\sqrt{-1-x}\sqrt{2} \right ) -\sqrt{-1-x} \right ) }{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63171, size = 180, normalized size = 2. \begin{align*} \frac{2 \,{\left (\sqrt{2}{\left (x - 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{x - 1}\right ) -{\left (x + 1\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}\right )}}{x - 1} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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