Optimal. Leaf size=48 \[ \frac {2}{3} \tanh (x) \sqrt {a \cosh ^3(x)}-\frac {2 i F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh ^3(x)}}{3 \cosh ^{\frac {3}{2}}(x)} \]
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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2635, 2641} \[ \frac {2}{3} \tanh (x) \sqrt {a \cosh ^3(x)}-\frac {2 i F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh ^3(x)}}{3 \cosh ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {a \cosh ^3(x)} \, dx &=\frac {\sqrt {a \cosh ^3(x)} \int \cosh ^{\frac {3}{2}}(x) \, dx}{\cosh ^{\frac {3}{2}}(x)}\\ &=\frac {2}{3} \sqrt {a \cosh ^3(x)} \tanh (x)+\frac {\sqrt {a \cosh ^3(x)} \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{3 \cosh ^{\frac {3}{2}}(x)}\\ &=-\frac {2 i \sqrt {a \cosh ^3(x)} F\left (\left .\frac {i x}{2}\right |2\right )}{3 \cosh ^{\frac {3}{2}}(x)}+\frac {2}{3} \sqrt {a \cosh ^3(x)} \tanh (x)\\ \end {align*}
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Mathematica [C] time = 0.05, size = 59, normalized size = 1.23 \[ \frac {2}{3} \sqrt {a \cosh ^3(x)} \left (\text {sech}^2(x) \sqrt {\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)^{3}}, x\right ) \]
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 \[ \int \sqrt {a \cosh \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\cosh ^{3}\relax (x )\right )}\, dx \]
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 \[ \int \sqrt {a \cosh \relax (x)^{3}}\,{d x} \]
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 \[ \int \sqrt {a\,{\mathrm {cosh}\relax (x)}^3} \,d x \]
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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