Optimal. Leaf size=13 \[ 2 \coth (x) \sqrt {\sinh (x) \tanh (x)} \]
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Rubi [A] time = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 2589
Rule 4398
Rule 4400
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*}
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Mathematica [A] time = 0.07, size = 13, normalized size = 1.00 \[ 2 \coth (x) \sqrt {\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 53, normalized size = 4.08 \[ \frac {2 \, \sqrt {\frac {1}{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )}}{\sqrt {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sinh \relax (x) \tanh \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 42, normalized size = 3.23 \[ \frac {\sqrt {2}\, \sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-x}}{1+{\mathrm e}^{2 x}}}\, \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.77, size = 35, normalized size = 2.69 \[ -\frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {e^{\left (-2 \, x\right )} + 1}} - \frac {\sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{\sqrt {e^{\left (-2 \, x\right )} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 35, normalized size = 2.69 \[ 2\,\mathrm {coth}\relax (x)\,\sqrt {-\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )\,\left ({\mathrm {e}}^{2\,x}-1\right )}\,\sqrt {\frac {1}{{\mathrm {e}}^{2\,x}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sinh {\relax (x )} \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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