Optimal. Leaf size=46 \[ 2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^3(x)}+2 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)} \]
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Rubi [A] time = 0.0342023, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2639} \[ 2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^3(x)}+2 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3768
Rule 3771
Rule 2639
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*}
Mathematica [A] time = 0.0187185, size = 36, normalized size = 0.78 \[ 2 \cosh (x) \sqrt{a \text{sech}^3(x)} \left (\sinh (x)+i \sqrt{\cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left ({\rm sech} \left (x\right ) \right ) ^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \operatorname{sech}\left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \operatorname{sech}\left (x\right )^{3}}, 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{a \operatorname{sech}^{3}{\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{a \operatorname{sech}\left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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