Optimal. Leaf size=13 \[ 2 \tanh (x) \sqrt{\cosh (x) \coth (x)} \]
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Rubi [A] time = 0.0700452, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4397, 4398, 4400, 2589} \[ 2 \tanh (x) \sqrt{\cosh (x) \coth (x)} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 4398
Rule 4400
Rule 2589
Rubi steps
\begin{align*} \int \sqrt{\text{csch}(x)+\sinh (x)} \, dx &=\int \sqrt{\cosh (x) \coth (x)} \, dx\\ &=\frac{\sqrt{\cosh (x) \coth (x)} \int \sqrt{-i \cosh (x) \coth (x)} \, dx}{\sqrt{-i \cosh (x) \coth (x)}}\\ &=\frac{\sqrt{\cosh (x) \coth (x)} \int \sqrt{\cosh (x)} \sqrt{-i \coth (x)} \, dx}{\sqrt{\cosh (x)} \sqrt{-i \coth (x)}}\\ &=2 \sqrt{\cosh (x) \coth (x)} \tanh (x)\\ \end{align*}
Mathematica [B] time = 0.0693421, size = 35, normalized size = 2.69 \[ \frac{2 \left (\sqrt [4]{-\sinh ^2(x)}-1\right ) \tanh (x) \sqrt{\cosh (x) \coth (x)}}{\sqrt [4]{-\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.121, 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 = 1.81146, size = 73, normalized size = 5.62 \begin{align*} \frac{\sqrt{2} e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{e^{\left (-x\right )} + 1} \sqrt{-e^{\left (-x\right )} + 1}} - \frac{\sqrt{2} e^{\left (-\frac{3}{2} \, x\right )}}{\sqrt{e^{\left (-x\right )} + 1} \sqrt{-e^{\left (-x\right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71093, 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 )} + \operatorname{csch}{\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{\operatorname{csch}\left (x\right ) + \sinh \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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