Optimal. Leaf size=21 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0120375, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3480, 206} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \sqrt{1+\coth (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\coth (x)}\right )\\ &=\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+\coth (x)}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0741134, size = 45, normalized size = 2.14 \[ \frac{(1+i) (\coth (x)+1)^{3/2} \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )}{(i (\coth (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 17, normalized size = 0.8 \begin{align*}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+{\rm coth} \left (x\right )}} \right ) \sqrt{2} \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 \sqrt{\coth \left (x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.31598, size = 181, normalized size = 8.62 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (2 \, \sqrt{2} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17684, size = 50, normalized size = 2.38 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left ({\left | 2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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