Optimal. Leaf size=13 \[ 2 \coth (x) \sqrt{\sinh (x) \tanh (x)} \]
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Rubi [A] time = 0.0504159, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4398, 4400, 2589} \[ 2 \coth (x) \sqrt{\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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Rule 4398
Rule 4400
Rule 2589
Rubi steps
\begin{align*} \int \sqrt{\sinh (x) \tanh (x)} \, dx &=\frac{\sqrt{\sinh (x) \tanh (x)} \int \sqrt{-\sinh (x) \tanh (x)} \, dx}{\sqrt{-\sinh (x) \tanh (x)}}\\ &=\frac{\sqrt{\sinh (x) \tanh (x)} \int \sqrt{i \sinh (x)} \sqrt{i \tanh (x)} \, dx}{\sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=2 \coth (x) \sqrt{\sinh (x) \tanh (x)}\\ \end{align*}
Mathematica [A] time = 0.0618208, size = 13, normalized size = 1. \[ 2 \coth (x) \sqrt{\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.145, size = 42, normalized size = 3.2 \begin{align*}{\frac{\sqrt{2} \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{2\,x}}-1}\sqrt{{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}{{\rm e}^{-x}}}{{{\rm e}^{2\,x}}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.65304, size = 47, normalized size = 3.62 \begin{align*} -\frac{\sqrt{2} e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{e^{\left (-2 \, x\right )} + 1}} - \frac{\sqrt{2} e^{\left (-\frac{3}{2} \, x\right )}}{\sqrt{e^{\left (-2 \, x\right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.22741, size = 201, normalized size = 15.46 \begin{align*} \frac{2 \, \sqrt{\frac{1}{2}}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}}{\sqrt{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\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{\sinh{\left (x \right )} \tanh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sinh \left (x\right ) \tanh \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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