Integrand size = 10, antiderivative size = 46 \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x) \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 \sinh (x) \cosh (x) \sqrt {a \text {sech}^3(x)}+2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)} \]
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Rule 2719
Rule 3853
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \text {sech}^3(x)} \int \text {sech}^{\frac {3}{2}}(x) \, dx}{\text {sech}^{\frac {3}{2}}(x)} \\ & = 2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)-\frac {\sqrt {a \text {sech}^3(x)} \int \frac {1}{\sqrt {\text {sech}(x)}} \, dx}{\text {sech}^{\frac {3}{2}}(x)} \\ & = 2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)-\left (\cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}\right ) \int \sqrt {\cosh (x)} \, dx \\ & = 2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 \cosh (x) \sqrt {a \text {sech}^3(x)} \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right ) \]
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\[\int \sqrt {a \operatorname {sech}\left (x \right )^{3}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \sqrt {a \text {sech}^3(x)} \, dx=2 \, \sqrt {2} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \]
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\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int \sqrt {a \operatorname {sech}^{3}{\left (x \right )}}\, dx \]
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\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{3}} \,d x } \]
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\[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{3}} \,d x } \]
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Timed out. \[ \int \sqrt {a \text {sech}^3(x)} \, dx=\int \sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^3}} \,d x \]
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