Integrand size = 13, antiderivative size = 65 \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3157, 2719} \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=-\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Rule 2719
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \cosh (x)+b \sinh (x)} \int \sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ & = -\frac {2 i E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.51 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.17 \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=\frac {b \left (-a^2+b^2\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cosh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right ) \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )+\sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )} \left (2 a^3 \sqrt {1-\frac {b^2}{a^2}} \cosh (x)-2 a \left (a^2-b^2\right ) \cosh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )+2 a^2 b \sqrt {1-\frac {b^2}{a^2}} \sinh (x)+a^2 b \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )-b^3 \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right )}{a b \sqrt {1-\frac {b^2}{a^2}} \sqrt {a \cosh (x)+b \sinh (x)} \sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )}} \]
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Time = 0.99 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.51
method | result | size |
default | \(-\frac {\sqrt {a^{2}-b^{2}}\, \cosh \left (x \right )}{\sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(33\) |
risch | \(\sqrt {2}\, \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{-x}}+\frac {\left (2 a -2 b \right ) \left (-\frac {2 \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )}{\left (a -b \right ) \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{x}}}+\frac {\left (-\frac {a +b}{a -b}+\frac {2 a +2 b}{a -b}\right ) \sqrt {-\left (a +b \right ) \left (a -b \right )}\, \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}\, \sqrt {-\frac {2 \left ({\mathrm e}^{x}-\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}\, \sqrt {-\frac {{\mathrm e}^{x} \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}\, \left (-\frac {2 \sqrt {-\left (a +b \right ) \left (a -b \right )}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}, \frac {\sqrt {2}}{2}\right )}{a +b}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}, \frac {\sqrt {2}}{2}\right )}{a +b}\right )}{\left (a +b \right ) \sqrt {a \,{\mathrm e}^{3 x}+{\mathrm e}^{3 x} b +a \,{\mathrm e}^{x}-{\mathrm e}^{x} b}}\right ) \sqrt {2}\, \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{-x}}\, \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{x}}}{2 a \,{\mathrm e}^{2 x}+2 b \,{\mathrm e}^{2 x}+2 a -2 b}\) | \(455\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=-2 \, \sqrt {2} \sqrt {a + b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - 2 \, \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )} \]
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\[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=\int \sqrt {a \cosh {\left (x \right )} + b \sinh {\left (x \right )}}\, dx \]
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\[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )} \,d x } \]
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\[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )} \,d x } \]
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Timed out. \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=\int \sqrt {a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )} \,d x \]
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