Optimal. Leaf size=69 \[ -\frac{10}{21} i a \cosh ^{\frac{3}{2}}(x) \text{EllipticF}\left (\frac{i x}{2},2\right ) \sqrt{a \text{sech}^3(x)}+\frac{10}{21} a \sinh (x) \sqrt{a \text{sech}^3(x)}+\frac{2}{7} a \tanh (x) \text{sech}(x) \sqrt{a \text{sech}^3(x)} \]
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Rubi [A] time = 0.0398334, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2641} \[ \frac{10}{21} a \sinh (x) \sqrt{a \text{sech}^3(x)}+\frac{2}{7} a \tanh (x) \text{sech}(x) \sqrt{a \text{sech}^3(x)}-\frac{10}{21} i a \cosh ^{\frac{3}{2}}(x) F\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \left (a \text{sech}^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \text{sech}^3(x)}\right ) \int \text{sech}^{\frac{9}{2}}(x) \, dx}{\text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{2}{7} a \text{sech}(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{\left (5 a \sqrt{a \text{sech}^3(x)}\right ) \int \text{sech}^{\frac{5}{2}}(x) \, dx}{7 \text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{10}{21} a \sqrt{a \text{sech}^3(x)} \sinh (x)+\frac{2}{7} a \text{sech}(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{\left (5 a \sqrt{a \text{sech}^3(x)}\right ) \int \sqrt{\text{sech}(x)} \, dx}{21 \text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{10}{21} a \sqrt{a \text{sech}^3(x)} \sinh (x)+\frac{2}{7} a \text{sech}(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{1}{21} \left (5 a \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{10}{21} i a \cosh ^{\frac{3}{2}}(x) F\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)}+\frac{10}{21} a \sqrt{a \text{sech}^3(x)} \sinh (x)+\frac{2}{7} a \text{sech}(x) \sqrt{a \text{sech}^3(x)} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0361048, size = 47, normalized size = 0.68 \[ \frac{2}{21} a \text{sech}(x) \sqrt{a \text{sech}^3(x)} \left (-5 i \cosh ^{\frac{5}{2}}(x) \text{EllipticF}\left (\frac{i x}{2},2\right )+3 \tanh (x)+5 \sinh (x) \cosh (x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm sech} \left (x\right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}}\, 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 \left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{3}{2}}\,{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}} 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 \left (a \operatorname{sech}^{3}{\left (x \right )}\right )^{\frac{3}{2}}\, 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 \left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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