Integrand size = 8, antiderivative size = 27 \[ \int \sqrt {a \cosh (x)} \, dx=-\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {\cosh (x)}} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2721, 2719} \[ \int \sqrt {a \cosh (x)} \, dx=-\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]
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Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \cosh (x)} \, dx=-\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {\cosh (x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(38)=76\).
Time = 0.52 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.37
method | result | size |
default | \(\frac {\sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}\, a \sqrt {2}\, \sqrt {-2 \cosh \left (\frac {x}{2}\right )^{2}+1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \left (\operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-2 \operatorname {EllipticE}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\) | \(118\) |
risch | \(\sqrt {2}\, \sqrt {a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}}+\frac {\left (-\frac {4 \left (a \,{\mathrm e}^{2 x}+a \right )}{a \sqrt {{\mathrm e}^{x} \left (a \,{\mathrm e}^{2 x}+a \right )}}+\frac {2 i \sqrt {-i \left ({\mathrm e}^{x}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{x}-i\right )}\, \sqrt {i {\mathrm e}^{x}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{x}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{x}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a \,{\mathrm e}^{3 x}+a \,{\mathrm e}^{x}}}\right ) \sqrt {2}\, \sqrt {a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}}\, \sqrt {a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{x}}}{2+2 \,{\mathrm e}^{2 x}}\) | \(171\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \sqrt {a \cosh (x)} \, dx=-2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - 2 \, \sqrt {a \cosh \left (x\right )} \]
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\[ \int \sqrt {a \cosh (x)} \, dx=\int \sqrt {a \cosh {\left (x \right )}}\, dx \]
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\[ \int \sqrt {a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right )} \,d x } \]
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\[ \int \sqrt {a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right )} \,d x } \]
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Timed out. \[ \int \sqrt {a \cosh (x)} \, dx=\int \sqrt {a\,\mathrm {cosh}\left (x\right )} \,d x \]
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