Optimal. Leaf size=48 \[ \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac {6 i a^2 E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{5 \sqrt {\cosh (x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2635, 2640, 2639} \[ \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac {6 i a^2 E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{5 \sqrt {\cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int (a \cosh (x))^{5/2} \, dx &=\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x)+\frac {1}{5} \left (3 a^2\right ) \int \sqrt {a \cosh (x)} \, dx\\ &=\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x)+\frac {\left (3 a^2 \sqrt {a \cosh (x)}\right ) \int \sqrt {\cosh (x)} \, dx}{5 \sqrt {\cosh (x)}}\\ &=-\frac {6 i a^2 \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{5 \sqrt {\cosh (x)}}+\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 41, normalized size = 0.85 \[ \frac {2 (a \cosh (x))^{5/2} \left (\sinh (x) \cosh ^{\frac {3}{2}}(x)-3 i E\left (\left .\frac {i x}{2}\right |2\right )\right )}{5 \cosh ^{\frac {5}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)} a^{2} \cosh \relax (x)^{2}, x\right ) \]
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 \[ \int \left (a \cosh \relax (x)\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 184, normalized size = 3.83 \[ \frac {\sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a^{3} \left (16 \left (\sinh ^{6}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+16 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+3 \sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-6 \sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right )}{5 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]
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 \[ \int \left (a \cosh \relax (x)\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a\,\mathrm {cosh}\relax (x)\right )}^{5/2} \,d x \]
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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