Optimal. Leaf size=27 \[ -\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2640, 2639} \[ -\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \sqrt {a \cosh (x)} \, dx &=\frac {\sqrt {a \cosh (x)} \int \sqrt {\cosh (x)} \, dx}{\sqrt {\cosh (x)}}\\ &=-\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {\cosh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)}, x\right ) \]
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 \[ \int \sqrt {a \cosh \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 118, normalized size = 4.37 \[ \frac {\sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-2 \EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]
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 \[ \int \sqrt {a \cosh \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {a\,\mathrm {cosh}\relax (x)} \,d x \]
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 \[ \int \sqrt {a \cosh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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