Optimal. Leaf size=33 \[ \frac{2 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x}{\sqrt{x+1}} \]
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Rubi [A] time = 0.0750883, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 37} \[ \frac{2 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 37
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{1}{\sqrt{1-x} x^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{1+x}}\\ &=\frac{2 \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x}{\sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.008271, size = 21, normalized size = 0.64 \[ \frac{2 \sqrt{1-\frac{1}{x^2}} x}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 22, normalized size = 0.7 \begin{align*} 2\,{\frac{-1+x}{\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 [A] time = 1.11657, size = 9, normalized size = 0.27 \begin{align*} 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5053, size = 50, normalized size = 1.52 \begin{align*} 2 \, \sqrt{x + 1} \sqrt{\frac{x - 1}{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 56.8934, size = 19, normalized size = 0.58 \begin{align*} \begin{cases} 2 \sqrt{x - 1} & \text{for}\: \left |{x}\right | > 1 \\2 i \sqrt{1 - x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15067, size = 18, normalized size = 0.55 \begin{align*} -2 i \, \sqrt{2} + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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