Optimal. Leaf size=46 \[ 2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853,
3856, 2719} \begin {gather*} 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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3853
Rule 3856
Rule 4208
Rubi steps
\begin {align*} \int \sqrt {a \text {sech}^3(x)} \, dx &=\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*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.78 \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.00, size = 0, normalized size = 0.00 \[\int \sqrt {a \mathrm {sech}\left (x \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 60, normalized size = 1.30 \begin {gather*} 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 ) \end {gather*}
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 \begin {gather*} \int \sqrt {a \operatorname {sech}^{3}{\left (x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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