Optimal. Leaf size=121 \[ \frac {26}{165} a^2 \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^3(x)}+\frac {78}{385} a^2 \sinh (x) \cosh (x) \sqrt {a \cosh ^3(x)}+\frac {26}{77} a^2 \tanh (x) \sqrt {a \cosh ^3(x)}-\frac {26 i a^2 F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh ^3(x)}}{77 \cosh ^{\frac {3}{2}}(x)}+\frac {2}{15} a^2 \sinh (x) \cosh ^5(x) \sqrt {a \cosh ^3(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2635, 2641} \[ \frac {2}{15} a^2 \sinh (x) \cosh ^5(x) \sqrt {a \cosh ^3(x)}+\frac {26}{165} a^2 \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^3(x)}+\frac {78}{385} a^2 \sinh (x) \cosh (x) \sqrt {a \cosh ^3(x)}+\frac {26}{77} a^2 \tanh (x) \sqrt {a \cosh ^3(x)}-\frac {26 i a^2 F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh ^3(x)}}{77 \cosh ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 3207
Rubi steps
\begin {align*} \int \left (a \cosh ^3(x)\right )^{5/2} \, dx &=\frac {\left (a^2 \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {15}{2}}(x) \, dx}{\cosh ^{\frac {3}{2}}(x)}\\ &=\frac {2}{15} a^2 \cosh ^5(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (13 a^2 \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {11}{2}}(x) \, dx}{15 \cosh ^{\frac {3}{2}}(x)}\\ &=\frac {26}{165} a^2 \cosh ^3(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{15} a^2 \cosh ^5(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (39 a^2 \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {7}{2}}(x) \, dx}{55 \cosh ^{\frac {3}{2}}(x)}\\ &=\frac {78}{385} a^2 \cosh (x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{165} a^2 \cosh ^3(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{15} a^2 \cosh ^5(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (39 a^2 \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {3}{2}}(x) \, dx}{77 \cosh ^{\frac {3}{2}}(x)}\\ &=\frac {78}{385} a^2 \cosh (x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{165} a^2 \cosh ^3(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{15} a^2 \cosh ^5(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{77} a^2 \sqrt {a \cosh ^3(x)} \tanh (x)+\frac {\left (13 a^2 \sqrt {a \cosh ^3(x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{77 \cosh ^{\frac {3}{2}}(x)}\\ &=-\frac {26 i a^2 \sqrt {a \cosh ^3(x)} F\left (\left .\frac {i x}{2}\right |2\right )}{77 \cosh ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cosh (x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{165} a^2 \cosh ^3(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{15} a^2 \cosh ^5(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{77} a^2 \sqrt {a \cosh ^3(x)} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 65, normalized size = 0.54 \[ \frac {a \left (a \cosh ^3(x)\right )^{3/2} \left ((15465 \sinh (x)+3657 \sinh (3 x)+749 \sinh (5 x)+77 \sinh (7 x)) \sqrt {\cosh (x)}-12480 i F\left (\left .\frac {i x}{2}\right |2\right )\right )}{36960 \cosh ^{\frac {9}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)^{3}} a^{2} \cosh \relax (x)^{6}, x\right ) \]
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 \[ \int \left (a \cosh \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int \left (a \left (\cosh ^{3}\relax (x )\right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int \left (a \cosh \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a\,{\mathrm {cosh}\relax (x)}^3\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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