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 = 0.0150689, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 2640
Rule 2639
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*}
Mathematica [A] time = 0.0092948, size = 27, normalized size = 1. \[ -\frac{2 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{\sqrt{\cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 118, normalized size = 4.4 \begin{align*}{a\sqrt{2}\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+1}\sqrt{- \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}} \left ({\it EllipticF} \left ( \cosh \left ({\frac{x}{2}} \right ) \sqrt{2},{\frac{\sqrt{2}}{2}} \right ) -2\,{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) \right ){\frac{1}{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \cosh \left (x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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