Optimal. Leaf size=121 \[ \frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}-\frac {26 i F\left (\left .\frac {i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \sinh (x) \cosh ^5(x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \sinh (x) \cosh ^3(x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}} \]
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Rubi [A] time = 0.06, 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 = {4123, 3769, 3771, 2641} \[ \frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \sinh (x) \cosh ^5(x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \sinh (x) \cosh ^3(x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}-\frac {26 i F\left (\left .\frac {i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx &=\frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (13 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {11}{2}}(x)} \, dx}{15 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (39 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {7}{2}}(x)} \, dx}{55 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (39 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)} \, dx}{77 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (13 \text {sech}^{\frac {3}{2}}(x)\right ) \int \sqrt {\text {sech}(x)} \, dx}{77 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {13 \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}\\ &=-\frac {26 i F\left (\left .\frac {i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 63, normalized size = 0.52 \[ \frac {\cosh (x) \sqrt {a \text {sech}^3(x)} \left (19122 \sinh (2 x)+4406 \sinh (4 x)+826 \sinh (6 x)+77 \sinh (8 x)-24960 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )\right )}{73920 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \operatorname {sech}\relax (x)^{3}}}{a^{3} \operatorname {sech}\relax (x)^{9}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \operatorname {sech}\relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\left (a \mathrm {sech}\relax (x )^{3}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (a \operatorname {sech}\relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\relax (x)}^3}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \operatorname {sech}^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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