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 = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 37
Rule 6176
Rule 6181
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.59, size = 18, normalized size = 0.55 \[ 2 \, \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 13, normalized size = 0.39 \[ -2 i \, \sqrt {2} + 2 \, \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 22, normalized size = 0.67 \[ \frac {-2+2 x}{\sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 7, normalized size = 0.21 \[ 2 \, \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 18, normalized size = 0.55 \[ 2\,\sqrt {\frac {x-1}{x+1}}\,\sqrt {x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.49, size = 19, normalized size = 0.58 \[ \begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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