Optimal. Leaf size=77 \[ -\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, 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 = {4208, 3854,
3856, 2719} \begin {gather*} \frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}-\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \sinh (x) \cosh ^2(x)}{9 a \sqrt {a \text {sech}^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3854
Rule 3856
Rule 4208
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx &=\frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \text {sech}^3(x)}}\\ &=\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}}+\frac {\left (7 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {5}{2}}(x)} \, dx}{9 a \sqrt {a \text {sech}^3(x)}}\\ &=\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}}+\frac {\left (7 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\text {sech}(x)}} \, dx}{15 a \sqrt {a \text {sech}^3(x)}}\\ &=\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}}+\frac {7 \int \sqrt {\cosh (x)} \, dx}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}\\ &=-\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 47, normalized size = 0.61 \begin {gather*} \frac {-\frac {84 i E\left (\left .\frac {i x}{2}\right |2\right )}{\cosh ^{\frac {3}{2}}(x)}+33 \sinh (x)+5 \sinh (3 x)}{90 a \sqrt {a \text {sech}^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.84, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \mathrm {sech}\left (x \right )^{3}\right )^{\frac {3}{2}}}\, dx\]
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 \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 = 0.16, size = 407, normalized size = 5.29 \begin {gather*} -\frac {672 \, \sqrt {2} {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - \sqrt {2} {\left (5 \, \cosh \left (x\right )^{10} + 50 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + 5 \, \sinh \left (x\right )^{10} + {\left (225 \, \cosh \left (x\right )^{2} + 43\right )} \sinh \left (x\right )^{8} + 43 \, \cosh \left (x\right )^{8} + 8 \, {\left (75 \, \cosh \left (x\right )^{3} + 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 2 \, {\left (525 \, \cosh \left (x\right )^{4} + 602 \, \cosh \left (x\right )^{2} - 149\right )} \sinh \left (x\right )^{6} - 298 \, \cosh \left (x\right )^{6} + 4 \, {\left (315 \, \cosh \left (x\right )^{5} + 602 \, \cosh \left (x\right )^{3} - 447 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (525 \, \cosh \left (x\right )^{6} + 1505 \, \cosh \left (x\right )^{4} - 2235 \, \cosh \left (x\right )^{2} - 187\right )} \sinh \left (x\right )^{4} - 374 \, \cosh \left (x\right )^{4} + 8 \, {\left (75 \, \cosh \left (x\right )^{7} + 301 \, \cosh \left (x\right )^{5} - 745 \, \cosh \left (x\right )^{3} - 187 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (225 \, \cosh \left (x\right )^{8} + 1204 \, \cosh \left (x\right )^{6} - 4470 \, \cosh \left (x\right )^{4} - 2244 \, \cosh \left (x\right )^{2} - 43\right )} \sinh \left (x\right )^{2} - 43 \, \cosh \left (x\right )^{2} + 2 \, {\left (25 \, \cosh \left (x\right )^{9} + 172 \, \cosh \left (x\right )^{7} - 894 \, \cosh \left (x\right )^{5} - 748 \, \cosh \left (x\right )^{3} - 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{720 \, {\left (a^{2} \cosh \left (x\right )^{5} + 5 \, a^{2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, a^{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, a^{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + a^{2} \sinh \left (x\right )^{5}\right )}} \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 \frac {1}{\left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {3}{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 \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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