Optimal. Leaf size=69 \[ \frac {10}{21} a \sinh (x) \sqrt {a \text {sech}^3(x)}+\frac {2}{7} a \tanh (x) \text {sech}(x) \sqrt {a \text {sech}^3(x)}-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 2641
Rule 3768
Rule 3771
Rule 4123
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*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.68 \[ \frac {2}{21} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \left (3 \tanh (x)-5 i \cosh ^{\frac {5}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )+5 \sinh (x) \cosh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \operatorname {sech}\relax (x)^{3}} a \operatorname {sech}\relax (x)^{3}, x\right ) \]
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 \[ \int \left (a \operatorname {sech}\relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \left (a \mathrm {sech}\relax (x )^{3}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int \left (a \operatorname {sech}\relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {a}{{\mathrm {cosh}\relax (x)}^3}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \operatorname {sech}^{3}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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