Optimal. Leaf size=121 \[ \frac {154}{195} i a^2 \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {154}{585} a^2 \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 7, 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*} \frac {154}{585} a^2 \tanh (x) \sqrt {a \text {sech}^3(x)}+\frac {2}{13} a^2 \tanh (x) \text {sech}^4(x) \sqrt {a \text {sech}^3(x)}+\frac {22}{117} a^2 \tanh (x) \text {sech}^2(x) \sqrt {a \text {sech}^3(x)}+\frac {154}{195} i a^2 \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+\frac {154}{195} a^2 \sinh (x) \cosh (x) \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 \left (a \text {sech}^3(x)\right )^{5/2} \, dx &=\frac {\left (a^2 \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {15}{2}}(x) \, dx}{\text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (11 a^2 \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {11}{2}}(x) \, dx}{13 \text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (77 a^2 \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {7}{2}}(x) \, dx}{117 \text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {154}{585} a^2 \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (77 a^2 \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {3}{2}}(x) \, dx}{195 \text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {154}{585} a^2 \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x)-\frac {\left (77 a^2 \sqrt {a \text {sech}^3(x)}\right ) \int \frac {1}{\sqrt {\text {sech}(x)}} \, dx}{195 \text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {154}{585} a^2 \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x)-\frac {1}{195} \left (77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}\right ) \int \sqrt {\cosh (x)} \, dx\\ &=\frac {154}{195} i a^2 \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {154}{585} a^2 \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 63, normalized size = 0.52 \begin {gather*} \frac {2}{585} a \text {sech}(x) \left (a \text {sech}^3(x)\right )^{3/2} \left (231 i \cosh ^{\frac {11}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+55 \cosh (x) \sinh (x)+77 \cosh ^3(x) \sinh (x)+231 \cosh ^5(x) \sinh (x)+45 \tanh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.05, size = 0, normalized size = 0.00 \[\int \left (a \mathrm {sech}\left (x \right )^{3}\right )^{\frac {5}{2}}\, dx\]
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 \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 = 0.11, size = 1382, normalized size = 11.42 \begin {gather*} \text {Too large to display} \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 \left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {5}{2}}\, 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 = -0.01 \begin {gather*} \int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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