Optimal. Leaf size=48 \[ \frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)}-\frac {2 i a^2 \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 \sqrt {a \cosh (x)}} \]
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Rubi [A] time = 0.02, 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, 2642, 2641} \[ \frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)}-\frac {2 i a^2 \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 \sqrt {a \cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int (a \cosh (x))^{3/2} \, dx &=\frac {2}{3} a \sqrt {a \cosh (x)} \sinh (x)+\frac {1}{3} a^2 \int \frac {1}{\sqrt {a \cosh (x)}} \, dx\\ &=\frac {2}{3} a \sqrt {a \cosh (x)} \sinh (x)+\frac {\left (a^2 \sqrt {\cosh (x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{3 \sqrt {a \cosh (x)}}\\ &=-\frac {2 i a^2 \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 \sqrt {a \cosh (x)}}+\frac {2}{3} a \sqrt {a \cosh (x)} \sinh (x)\\ \end {align*}
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Mathematica [C] time = 0.06, size = 57, normalized size = 1.19 \[ \frac {2}{3} (a \cosh (x))^{3/2} \left (\text {sech}^2(x) \sqrt {\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)} a \cosh \relax (x), x\right ) \]
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 \[ \int \left (a \cosh \relax (x)\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 130, normalized size = 2.71 \[ \frac {\sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a^{2} \left (8 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+\sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right )}{3 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]
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 \[ \int \left (a \cosh \relax (x)\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]
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 \[ \int \left (a \cosh {\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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