Optimal. Leaf size=13 \[ 2 \tanh (x) \sqrt {\cosh (x) \coth (x)} \]
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Rubi [A] time = 0.07, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 2589
Rule 4397
Rule 4398
Rule 4400
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*}
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Mathematica [B] time = 0.07, 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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fricas [B] time = 0.41, size = 55, normalized size = 4.23 \[ \frac {2 \, \sqrt {\frac {1}{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )}}{\sqrt {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)}} \]
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 \[ \int \sqrt {\operatorname {csch}\relax (x) + \sinh \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 42, normalized size = 3.23 \[ \frac {\sqrt {2}\, \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-x}}{{\mathrm e}^{2 x}-1}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 54, normalized size = 4.15 \[ \frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} - \frac {\sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 13, normalized size = 1.00 \[ 2\,\mathrm {tanh}\relax (x)\,\sqrt {\mathrm {sinh}\relax (x)+\frac {1}{\mathrm {sinh}\relax (x)}} \]
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 \[ \int \sqrt {\sinh {\relax (x )} + \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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