Integrand size = 10, antiderivative size = 54 \[ \int \sqrt {\sinh (a+b x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {i \sinh (a+b x)}} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2721, 2719} \[ \int \sqrt {\sinh (a+b x)} \, dx=-\frac {2 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)}} \]
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Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\sinh (a+b x)} \int \sqrt {i \sinh (a+b x)} \, dx}{\sqrt {i \sinh (a+b x)}} \\ & = -\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {i \sinh (a+b x)}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \sqrt {\sinh (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (a+b x)\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{b \sqrt {\sinh (a+b x)}} \]
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Time = 0.89 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.00
method | result | size |
default | \(\frac {\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(108\) |
risch | \(\frac {\sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}}{b}-\frac {\left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}\, \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}\) | \(210\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int \sqrt {\sinh (a+b x)} \, dx=-\frac {2 \, {\left (\sqrt {2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right ) + \sqrt {\sinh \left (b x + a\right )}\right )}}{b} \]
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\[ \int \sqrt {\sinh (a+b x)} \, dx=\int \sqrt {\sinh {\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {\sinh (a+b x)} \, dx=\int { \sqrt {\sinh \left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {\sinh (a+b x)} \, dx=\int { \sqrt {\sinh \left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\sinh (a+b x)} \, dx=\int \sqrt {\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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